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Statistics By Jim

Making statistics intuitive

Failing to Reject the Null Hypothesis

By Jim Frost 69 Comments

Failing to reject the null hypothesis is an odd way to state that the results of your hypothesis test are not statistically significant. Why the peculiar phrasing? “Fail to reject” sounds like one of those double negatives that writing classes taught you to avoid. What does it mean exactly? There’s an excellent reason for the odd wording!

In this post, learn what it means when you fail to reject the null hypothesis and why that’s the correct wording. While accepting the null hypothesis sounds more straightforward, it is not statistically correct!

Before proceeding, let’s recap some necessary information. In all statistical hypothesis tests, you have the following two hypotheses:

• The null hypothesis states that there is no effect or relationship between the variables.
• The alternative hypothesis states the effect or relationship exists.

We assume that the null hypothesis is correct until we have enough evidence to suggest otherwise.

After you perform a hypothesis test, there are only two possible outcomes.

• When your p-value is greater than your significance level, you fail to reject the null hypothesis. Your results are not significant. You’ll learn more about interpreting this outcome later in this post.

Related posts : Hypothesis Testing Overview and The Null Hypothesis

Why Don’t Statisticians Accept the Null Hypothesis?

To understand why we don’t accept the null, consider the fact that you can’t prove a negative. A lack of evidence only means that you haven’t proven that something exists. It does not prove that something doesn’t exist. It might exist, but your study missed it. That’s a huge difference and it is the reason for the convoluted wording. Let’s look at several analogies.

Species Presumed to be Extinct

Lack of proof doesn’t represent proof that something doesn’t exist!

Criminal Trials

Perhaps the prosecutor conducted a shoddy investigation and missed clues? Or, the defendant successfully covered his tracks? Consequently, the verdict in these cases is “not guilty.” That judgment doesn’t say the defendant is proven innocent, just that there wasn’t enough evidence to move the jury from the default assumption of innocence.

Hypothesis Tests

The hypothesis test assesses the evidence in your sample. If your test fails to detect an effect, it’s not proof that the effect doesn’t exist. It just means your sample contained an insufficient amount of evidence to conclude that it exists. Like the species that were presumed extinct, or the prosecutor who missed clues, the effect might exist in the overall population but not in your particular sample. Consequently, the test results fail to reject the null hypothesis, which is analogous to a “not guilty” verdict in a trial. There just wasn’t enough evidence to move the hypothesis test from the default position that the null is true.

The critical point across these analogies is that a lack of evidence does not prove something does not exist—just that you didn’t find it in your specific investigation. Hence, you never accept the null hypothesis.

Related post : The Significance Level as an Evidentiary Standard

What Does Fail to Reject the Null Hypothesis Mean?

Accepting the null hypothesis would indicate that you’ve proven an effect doesn’t exist. As you’ve seen, that’s not the case at all. You can’t prove a negative! Instead, the strength of your evidence falls short of being able to reject the null. Consequently, we fail to reject it.

Failing to reject the null indicates that our sample did not provide sufficient evidence to conclude that the effect exists. However, at the same time, that lack of evidence doesn’t prove that the effect does not exist. Capturing all that information leads to the convoluted wording!

What are the possible implications of failing to reject the null hypothesis? Let’s work through them.

First, it is possible that the effect truly doesn’t exist in the population, which is why your hypothesis test didn’t detect it in the sample. Makes sense, right? While that is one possibility, it doesn’t end there.

Another possibility is that the effect exists in the population, but the test didn’t detect it for a variety of reasons. These reasons include the following:

• The sample size was too small to detect the effect.
• The variability in the data was too high. The effect exists, but the noise in your data swamped the signal (effect).
• By chance, you collected a fluky sample. When dealing with random samples, chance always plays a role in the results. The luck of the draw might have caused your sample not to reflect an effect that exists in the population.

Notice how studies that collect a small amount of data or low-quality data are likely to miss an effect that exists? These studies had inadequate statistical power to detect the effect. We certainly don’t want to take results from low-quality studies as proof that something doesn’t exist!

However, failing to detect an effect does not necessarily mean a study is low-quality. Random chance in the sampling process can work against even the best research projects!

If you’re learning about hypothesis testing and like the approach I use in my blog, check out my eBook!

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Reader Interactions

May 8, 2024 at 9:08 am

Thank you very much for explaining the topic. It brings clarity and makes statistics very simple and interesting. Its helping me in the field of Medical Research.

February 26, 2024 at 7:54 pm

Hi Jim, My question is that can I reverse Null hyposthesis and start with Null: µ1 ≠ µ2 ? Then, if I can reject Null, I will end up with µ1=µ2 for mean comparison and this what I am looking for. But isn’t this cheating?

February 26, 2024 at 11:41 pm

That can be done but it requires you to revamp the entire test. Keep in mind that the reason you normally start out with the null equating to no relationship is because the researchers typically want to prove that a relationship or effect exists. This format forces the researchers to collect a substantial amount of high quality data to have a chance at demonstrating that an effect exists. If they collect a small sample and/or poor quality (e.g., noisy or imprecise), then the results default back to the null stating that no effect exists. So, they have to collect good data and work hard to get findings that suggest the effect exists.

There are tests that flip it around as you suggest where the null states that a relationship does exist. For example, researchers perform an equivalency test when they want to show that there is no difference. That the groups are equal. The test is designed such that it requires a good sample size and high quality data to have a chance at proving equivalency. If they have a small sample size and/or poor quality data, the results default back to the groups being unequal, which is not what they want to show.

So, choose the null hypothesis and corresponding analysis based on what you hope to find. Choose the null hypothesis that forces you to work hard to reject it and get the results that you want. It forces you to collect better evidence to make your case and the results default back to what you don’t want if you do a poor job.

I hope that makes sense!

October 13, 2023 at 5:10 am

Really appreciate how you have been able to explain something difficult in very simple terms. Also covering why you can’t accept a null hypothesis – something which I think is frequently missed. Thank you, Jim.

February 22, 2022 at 11:18 am

Hi Jim, I really appreciate your blog, making difficult things sound simple is a great gift.

I have a doubt about the p-value. You said there are two options when it comes to hypothesis tests results . Reject or failing to reject the null, depending on the p-value and your significant level.

But… a P-value of 0,001 means a stronger evidence than a P-value of 0,01? ( both with a significant level of 5%. Or It doesn`t matter, and just every p-Value under your significant level means the same burden of evidence against the null?

I hope I made my point clear. Thanks a lot for your time.

February 23, 2022 at 9:06 pm

There are different schools of thought about this question. The traditional approach is clear cut. Your results are statistically significance when your p-value is less than or equal to your significance level. When the p-value is greater than the significance level, your results are not significant.

However, as you point out, lower p-values indicate stronger evidence against the null hypothesis. I write about this aspect of p-values in several articles, interpreting p-values (near the end) and p-values and reproducibility .

Personally, I consider both aspects. P-values near 0.05 provide weak evidence. Consequently, I’d be willing to say that p-values less than or equal to 0.05 are statistically significant, but when they’re near 0.05, I’d consider it as a preliminary result that requires more research. However, if the p-value is less 0.01, or even better 0.001, then that’s much stronger evidence and I’ll give those results more weight in my evaluation.

If you read those two articles, I think you’ll see what I mean.

January 1, 2022 at 6:00 pm

HI, I have a quick question that you may be able to help me with. I am using SPSS and carrying out a Mann W U Test it says to retain the null hypothesis. The hypothesis is that males are faster than women at completing a task. So is that saying that they are or are not

January 1, 2022 at 8:17 pm

In that case, your sample data provides insufficient evidence to conclude that males are faster. The results do not prove that males and females are the same speed. You just don’t have enough evidence to say males are faster. In this post, I cover the reasons why you can’t prove the null is true.

November 23, 2021 at 5:36 pm

What if I have to prove in my hypothesis that there shouldn’t be any affect of treatment on patients? Can I say that if my null hypothesis is accepted i have got my results (no effect)? I am confused what to do in this situation. As for null hypothesis we always have to write it with some type of equality. What if I want my result to be what i have stated in null hypothesis i.e. no effect? How to write statements in this case? I am using non parametric test, Mann whitney u test

November 27, 2021 at 4:56 pm

You need to perform an equivalence test, which is a special type of procedure when you want to prove that the results are equal. The problem with a regular hypothesis test is that when you fail to reject the null, you’re not proving that they the outcomes are equal. You can fail to reject the null thanks to a small sample size, noisy data, or a small effect size even when the outcomes are truly different at the population level. An equivalence test sets things up so you need strong evidence to really show that two outcomes are equal.

Unfortunately, I don’t have any content for equivalence testing at this point, but you can read an article about it at Wikipedia: Equivalence Test .

August 13, 2021 at 9:41 pm

Great explanation and great analogies! Thanks.

August 11, 2021 at 2:02 am

I got problems with analysis. I did wound healing experiments with drugs treatment (total 9 groups). When I do the 2-way ANOVA in excel, I got the significant results in sample (Drug Treatment) and columns (Day, Timeline) . But I did not get the significantly results in interactions. Can I still reject the null hypothesis and continue the post-hoc test?

Thank you very much.

June 13, 2021 at 4:51 am

Hi Jim, There are so many books covering maths/programming related to statistics/DS, but may be hardly any book to develop an intuitive understanding. Thanks to you for filling up that gap. After statistics, hypothesis-testing, regression, will it be possible for you to write such books on more topics in DS such as trees, deep-learning etc.

I recently started with reading your book on hypothesis testing (just finished the first chapter). I have a question w.r.t the fuel cost example (from first chapter), where a random sample of 25 families (with sample mean 330.6) is taken. To do the hypothesis testing here, we are taking a sampling distribution with a mean of 260. Then based on the p-value and significance level, we find whether to reject or accept the null hypothesis. The entire decision (to accept or reject the null hypothesis) is based on the sampling distribution about which i have the following questions : a) we are assuming that the sampling distribution is normally distributed. what if it has some other distribution, how can we find that ? b) We have assumed that the sampling distribution is normally distributed and then further assumed that its mean is 260 (as required for the hypothesis testing). But we need the standard deviation as well to define the normal distribution, can you please let me know how do we find the standard deviation for the sampling distribution ? Thanks.

April 24, 2021 at 2:25 pm

Maybe its the idea of “Innocent until proven guilty”? Your Null assume the person is not guilty, and your alternative assumes the person is guilty, only when you have enough evidence (finding statistical significance P0.05 you have failed to reject null hypothesis, null stands,implying the person is not guilty. Or, the person remain innocent.. Correct me if you think it’s wrong but this is the way I interpreted.

April 25, 2021 at 5:10 pm

I used the courtroom/trial analogy within this post. Read that for more details. I’d agree with your general take on the issue except when you have enough evidence you actually reject the null, which in the trial means the defendant is found guilty.

April 17, 2021 at 6:10 am

Can regression analysis be done using 5 companies variables for predicting working capital management and profitability positive/negative relationship?

Also, does null hypothesis rejecting means whatsoever is stated in null hypothesis that is false proved through regression analysis?

I have very less knowledge about regression analysis. Please help me, Sir. As I have my project report due on next week. Thanks in advance!

April 18, 2021 at 10:48 pm

Hi Ahmed, yes, regression analysis can be used for the scenario you describe as long as you have the required data.

For more about the null hypothesis in relation to regression analysis, read my post about regression coefficients and their p-values . I describe the null hypothesis in it.

January 26, 2021 at 7:32 pm

With regards to the legal example above. While your explanation makes sense when simplified to this statistical level, from a legal perspective it is not correct. The presumption of innocence means one does not need to be proven innocent. They are innocent. The onus of proof lies with proving they are guilty. So if you can’t prove someones guilt then in fact you must accept the null hypothesis that they are innocent. It’s not a statistical test so a little bit misleading using it an example, although I see why you would.

If it were a statistical test, then we would probably be rather paranoid that everyone is a murderer but they just haven’t been proven to be one yet.

Great article though, a nice simple and thoughtout explanation.

January 26, 2021 at 9:11 pm

It seems like you misread my post. The hypothesis testing/legal analogy is very strong both in making the case and in the result.

In hypothesis testing, the data have to show beyond a reasonable doubt that the alternative hypothesis is true. In a court case, the prosecutor has to present sufficient evidence to show beyond a reasonable doubt that the defendant is guilty.

In terms of the test/case results. When the evidence (data) is insufficient, you fail to reject the null hypothesis but you do not conclude that the data proves the null is true. In a legal case that has insufficient evidence, the jury finds the defendant to be “not guilty” but they do not say that s/he is proven innocent. To your point specifically, it is not accurate to say that “not guilty” is the same as “proven innocent.”

It’s a very strong parallel.

January 9, 2021 at 11:45 am

Just a question, in my research on hypotheses for an assignment, I am finding it difficult to find an exact definition for a hypothesis itself. I know the defintion, but I’m looking for a citable explanation, any ideas?

January 10, 2021 at 1:37 am

To be clear, do you need to come up with a statistical hypothesis? That’s one where you’ll use a particular statistical hypothesis test. If so, I’ll need to know more about what you’re studying, your variables, and the type of hypothesis test you plan to use.

There are also scientific hypotheses that you’ll state in your proposals, study papers, etc. Those are different from statistical hypotheses (although related). However, those are very study area specific and I don’t cover those types on this blog because this is a statistical blog. But, if it’s a statistical hypothesis for a hypothesis test, then let me know the information I mention above and I can help you out!

November 7, 2020 at 8:33 am

Hi, good read, I’m kind of a novice here, so I’m trying to write a research paper, and I’m trying to make a hypothesis. however looking at the literature, there are contradicting results.

researcher A found that there is relationship between X and Y

however, researcher B found that there is no relationship between X and Y

therefore, what is the null hypothesis between X and y? do we choose what we assumed to be correct for our study? or is is somehow related to the alternative hypothesis? I’m confused.

thank you very much for the help.

November 8, 2020 at 12:07 am

Hypotheses for a statistical test are different than a researcher’s hypothesis. When you’re constructing the statistical hypothesis, you don’t need to consider what other researchers have found. Instead, you construct them so that the test only produces statistically significant results (rejecting the null) when your data provides strong evidence. I talk about that process in this post.

Typically, researchers are hoping to establish that an effect or relationship exists. Consequently, the null and alternative hypotheses are typically the following:

Null: The effect or relationship doesn’t not exist. Alternative: The effect or relationship does exist.

However, if you’re hoping to prove that there is no effect or no relationship, you then need to flip those hypotheses and use a special test, such as an equivalences test.

So, there’s no need to consider what researchers have found but instead what you’re looking for. In most cases, you are looking for an effect/relationship, so you’d go with the hypotheses as I show them above.

I hope that helps!

October 22, 2020 at 6:13 pm

Great, deep detailed answer. Appreciated!

September 16, 2020 at 12:03 pm

Thank you for explaining it too clearly. I have the following situation with a Box Bohnken design of three levels and three factors for multiple responses. F-value for second order model is not significant (failing to reject null hypothesis, p-value > 0.05) but, lack of fit of the model is not significant. What can you suggest me about statistical analysis?

September 17, 2020 at 2:42 am

Are your first order effects significant?

You want the lack of fit to be nonsignificant. If it’s significant, that means the model doesn’t fit the data well. So, you’re good there! 🙂

September 14, 2020 at 5:18 pm

thank you for all the explicit explanation on the subject.

However, i still got a question about “accepting the null hypothesis”. from textbook, the p-value is the probability that a statistic would take a value that is as extreme as or more extreme than that actually observed.

so, that’s why when p<0.01 we reject the null hypothesis, because it's too rare (p0.05, i can understand that for most cases we cannot accept the null, for example, if p=0.5, it means that the probability to get a statistic from the distribution is 0.5, which is totally random.

But how about when the p is very close to 1, like p=0.95, or p=0.99999999, can’t we say that the probability that the statistic is not from this distribution is less than 0.05, | or in another way, the probability that the statistic is from the distribution is almost 1. can’t we accept the null in such circumstance?

September 11, 2020 at 12:14 pm

Wow! This is beautifully explained. “Lack of proof doesn’t represent proof that something doesn’t exist!”. This kinda, hit me with such force. Can I then, use the same analogy for many other things in life? LOL! 🙂

H0 = God does not exist; H1 = God does exist; WE fail to reject H0 as there is no evidence.

Thank you sir, this has answered many of my questions, statistically speaking! No pun intended with the above.

September 11, 2020 at 4:58 pm

Hi, LOL, I’m glad it had such meaning for you! I’ll leave the determination about the existence of god up to each person, but in general, yes, I think statistical thinking can be helpful when applied to real life. It is important to realize that lack of proof truly is not proof that something doesn’t exist. But, I also consider other statistical concepts, such as confounders and sampling methodology, to be useful keeping in mind when I’m considering everyday life stuff–even when I’m not statistically analyzing it. Those concepts are generally helpful when trying to figure out what is going on in your life! Are there other alternative explanations? Is what you’re perceiving likely to be biased by something that’s affecting the “data” you can observe? Am I drawing a conclusion based on a large or small sample? How strong is the evidence?

A lot of those concepts are great considerations even when you’re just informally assessing and draw conclusions about things happening in your daily life.

August 13, 2020 at 12:04 am

Dear Jim, thanks for clarifying. absolutely, now it makes sense. the topic is murky but it is good to have your guidance, and be clear. I have not come across an instructor as clear in explaining as you do. Appreciate your direction. Thanks a lot, Geetanjali

August 15, 2020 at 3:48 pm

Hi Geetanjali,

I’m glad my website is helpful! That makes my day hearing that. Thanks so much for writing!

August 12, 2020 at 9:37 am

Hi Jim. I am doing data analyis for my masters thesis and my hypothesis testings were insignificant. And I am ok with that. But there is something bothering me. It is the low reliabilities of the 4-Items sub-scales (.55, .68, .75), though the overall alpha is good (.85). I just wonder if it is affecting my hypothesis testings.

August 11, 2020 at 9:23 pm

Thank you sir for replying, yes sir we it’s a RCT study.. where we did within and between the groups analysis and found p>0.05 in between the groups using Mann Whitney U test. So in such cases if the results comes like this we need to Mention that we failed reject the null hypothesis? Is that correct? Whether it tells that the study is inefficient as we couldn’t accept the alternative hypothesis. Thanks is advance.

August 11, 2020 at 9:43 pm

Hi Saumya, ah, this becomes clearer. When ask statistical questions, please be sure to include all relevant information because the details are extremely important. I didn’t know it was an RCT with a treatment and control group. Yes, given that your p-value is greater than your significance level, you fail to reject the null hypothesis. The results are not significant. The experiment provides insufficient evidence to conclude that the outcome in the treatment group is different than the control group.

By the way, you never accept the alternative hypothesis (or the null). The two options are to either reject the null or fail to reject the null. In your case, you fail to reject the null hypothesis.

I hope this helps!

August 11, 2020 at 9:41 am

Sir, p value is0.05, by which we interpret that both the groups are equally effective. In this case I had to reject the alternative hypothesis/ failed to reject null hypothessis.

August 11, 2020 at 12:37 am

sir, within the group analysis the p value for both the groups is significant (p0.05, by which we interpret that though both the treatments are effective, there in no difference between the efficacy of one over the other.. in other words.. no intervention is superior and both are equally effective.

August 11, 2020 at 2:45 pm

Thanks for the additional details. If I understand correctly, there were separate analyses before that determined each treatment had a statistically significance effect. However, when you compare the two treatments, there difference between them is not statistically significant.

If that’s the case, the interpretation is fairly straightforward. You have evidence that suggests that both treatments are effective. However, you don’t have evidence to conclude that one is better than the other.

August 10, 2020 at 9:26 am

Hi thank you for a wonderful explanation. I have a doubt: My Null hypothesis says: no significant difference between the effect fo A and B treatment Alternative hypothesis: there will be significant difference between the effect of A and B treatment. and my results show that i fail to reject null hypothesis.. Both the treatments were effective, but not significant difference.. how do I interpret this?

August 10, 2020 at 1:32 pm

First, I need to ask you a question. If your p-value is not significant, and so you fail to reject the null, why do you say that the treatment is effective? I can answer you question better after knowing the reason you say that. Thanks!

August 9, 2020 at 9:40 am

Dear Jim, thanks for making stats much more understandable and answering all question so painstakingly. I understand the following on p value and null. If our sample yields a p value of .01, it means that that there is a 1% probability that our kind of sample exists in the population. that is a rare event. So why shouldn’t we accept the HO as the probability of our event was v rare. Pls can you correct me. Thanks, G

August 10, 2020 at 1:53 pm

That’s a great question! They key thing to remember is that p-values are a conditional probability. P-value calculations assume that the null hypothesis is true. So, a p-value of 0.01 indicates that there is a 1% probability of observing your sample results, or more extreme, *IF* the null hypothesis is true.

The kicker is that we don’t whether the null is true or not. But, using this process does limit the likelihood of a false positive to your significance level (alpha). But, we don’t know whether the null is true and you had an unusual sample or whether the null is false. Usually, with a p-value of 0.01, we’d reject the null and conclude it is false.

I hope that answered your question. This topic can be murky and I wasn’t quite clear which part you needed clarification.

August 4, 2020 at 11:16 pm

Thank you for the wonderful explanation. However, I was just curious to know that what if in a particular test, we get a p-value less than the level of significance, leading to evidence against null hypothesis. Is there any possibility that our interpretation of population effect might be wrong due to randomness of samples? Also, how do we conclude whether the evidence is enough for our alternate hypothesis?

August 4, 2020 at 11:55 pm

Hi Abhilash,

Yes, unfortunately, when you’re working with samples, there’s always the possibility that random chance will cause your sample to not represent the population. For information about these errors, read my post about the types of errors in hypothesis testing .

In hypothesis testing, you determine whether your evidence is strong enough to reject the null. You don’t accept the alternative hypothesis. I cover that in my post about interpreting p-values .

August 1, 2020 at 3:50 pm

Hi, I am trying to interpret this phenomenon after my research. The null hypothesis states that “The use of combined drugs A and B does not lower blood pressure when compared to if drug A or B is used singularly”

The alternate hypothesis states: The use of combined drugs A and B lower blood pressure compared to if drug A or B is used singularly.

At the end of the study, majority of the people did not actually combine drugs A and B, rather indicated they either used drug A or drug B but not a combination. I am finding it very difficult to explain this outcome more so that it is a descriptive research. Please how do I go about this? Thanks a lot

June 22, 2020 at 10:01 am

What confuses me is how we set/determine the null hypothesis? For example stating that two sets of data are either no different or have no relationship will give completely different outcomes, so which is correct? Is the null that they are different or the same?

June 22, 2020 at 2:16 pm

Typically, the null states there is no effect/no relationship. That’s true for 99% of hypothesis tests. However, there are some equivalence tests where you are trying to prove that the groups are equal. In that case, the null hypothesis states that groups are not equal.

The null hypothesis is typically what you *don’t* want to find. You have to work hard, design a good experiment, collect good data, and end up with sufficient evidence to favor the alternative hypothesis. Usually in an experiment you want to find an effect. So, usually the null states there is no effect and you have get good evidence to reject that notion.

However, there are a few tests where you actually want to prove something is equal, so you need the null to state that they’re not equal in those cases and then do all the hard work and gather good data to suggest that they are equal. Basically, set up the hypothesis so it takes a good experiment and solid evidence to be able to reject the null and favor the hypothesis that you’re hoping is true.

June 5, 2020 at 11:54 am

Thank you for the explanation. I have one question that. If Null hypothesis is failed to reject than is possible to interpret the analysis further?

June 5, 2020 at 7:36 pm

Hi Mottakin,

Typically, if your result is that you fail to reject the null hypothesis there’s not much further interpretation. You don’t want to be in a situation where you’re endlessly trying new things on a quest for obtaining significant results. That’s data mining.

May 25, 2020 at 7:55 am

I hope all is well. I am enjoying your blog. I am not a statistician, however, I use statistical formulae to provide insight on the direction in which data is going. I have used both the regression analysis and a T-Test. I know that both use a null hypothesis and an alternative hypothesis. Could you please clarity the difference between a regression analysis and a T-Test? Are there conditions where one is a better option than the other?

May 26, 2020 at 9:18 pm

t-Tests compare the means of one or two groups. Regression analysis typically describes the relationships between a set of independent variables and the dependent variables. Interestingly, you can actually use regression analysis to perform a t-test. However, that would be overkill. If you just want to compare the means of one or two groups, use a t-test. Read my post about performing t-tests in Excel to see what they can do. If you have a more complex model than just comparing one or two means, regression might be the way to go. Read my post about when to use regression analysis .

May 12, 2020 at 5:45 pm

This article is really enlightening but there is still some darkness looming around. I see that low p-values mean strong evidence against null hypothesis and finding such a sample is highly unlikely when null hypothesis is true. So , is it OK to say that when p-value is 0.01 , it was very unlikely to have found such a sample but we still found it and hence finding such a sample has not occurred just by chance which leads towards rejection of null hypothesis.

May 12, 2020 at 11:16 pm

That’s mostly correct. I wouldn’t say, “has not occurred by chance.” So, when you get a very low p-value it does mean that you are unlikely to obtain that sample if the null is true. However, once you obtain that result, you don’t know for sure which of the two occurred:

• The effect exists in the population.
• Random chance gave you an unusual sample (i.e., Type I error).

You really don’t know for sure. However, by the decision making results you set about the strength of evidence required to reject the null, you conclude that the effect exists. Just always be aware that it could be a false positive.

That’s all a long way of saying that your sample was unlikely to occur by chance if the null is true.

April 29, 2020 at 11:59 am

Why do we consult the statistical tables to find out the critical values of our test statistics?

April 30, 2020 at 5:05 pm

Statistical tables started back in the “olden days” when computers didn’t exist. You’d calculate the test statistic value for your sample. Then, you’d look in the appropriate table and using the degrees of freedom for your design and find the critical values for the test statistic. If the value of your test statistics exceeded the critical value, your results were statistically significant.

With powerful and readily available computers, researchers could analyze their data and calculate the p-values and compare them directly to the significance level.

I hope that answers your question!

April 15, 2020 at 10:12 am

If we are not able to reject the null hypothesis. What could be the solution?

April 16, 2020 at 11:13 pm

Hi Shazzad,

The first thing to recognize is that failing to reject the null hypothesis might not be an error. If the null hypothesis is false, then the correct outcome is failing to reject the null.

However, if the null hypothesis is false and you fail to reject, it is a type II error, or a false negative. Read my post about types of errors in hypothesis tests for more information.

This type of error can occur for a variety of reasons, including the following:

• Fluky sample. When working with random samples, random error can cause anomalous results purely by chance.
• Sample is too small. Perhaps the sample was too small, which means the test didn’t have enough statistical power to detect the difference.
• Problematic data or sampling methodology. There could be a problem with how you collected the data or your sampling methodology.

There are various other possibilities, but those are several common problems.

April 14, 2020 at 12:19 pm

Thank you so much for this article! I am taking my first Statistics class in college and I have one question about this.

I understand that the default position is that the null is correct, and you explained that (just like a court case), the sample evidence must EXCEED the “evidentiary standard” (which is the significance level) to conclude that an effect/relationship exists. And, if an effect/relationship exists, that means that it’s the alternative hypothesis that “wins” (not sure if that’s the correct way of wording it, but I’m trying to make this as simple as possible in my head!).

But what I don’t understand is that if the P-value is GREATER than the significance value, we fail to reject the null….because shouldn’t a higher P-value, mean that our sample evidence EXCEEDS the evidentiary standard (aka the significance level), and therefore an effect/relationship exists? In my mind it would make more sense to reject the null, because our P-value is higher and therefore we have enough evidence to reject the null.

I hope I worded this in a way that makes sense. Thank you in advance!

April 14, 2020 at 10:42 pm

That’s a great question. The key thing to remember is that higher p-values correspond to weaker evidence against the null hypothesis. A high p-value indicates that your sample is likely (high probability = high p-value) if the null hypothesis is true. Conversely, low p-values represent stronger evidence against the null. You were unlikely (low probability = low p-value) to have collect a sample with the measured characteristics if the null is true.

So, there is negative correlation between p-values and strength of evidence against the null hypothesis. Low p-values indicate stronger evidence. Higher p-value represent weaker evidence.

In a nutshell, you reject the null hypothesis with a low p-value because it indicates your sample data are unusual if the null is true. When it’s unusual enough, you reject the null.

March 5, 2020 at 11:10 am

There is something I am confused about. If our significance level is .05 and our resulting p-value is .02 (thus the strength of our evidence is strong enough to reject the null hypothesis), do we state that we reject the null hypothesis with 95% confidence or 98% confidence?

My guess is our confidence level is 95% since or alpha was .05. But if the strength of our evidence is 98%, why wouldn’t we use that as our stated confidence in our results?

March 5, 2020 at 4:19 pm

Hi Michael,

You’d state that you can reject the null at a significance level of 5% or conversely at the 95% confidence level. A key reason is to avoid cherry picking your results. In other words, you don’t want to choose the significance level based on your results.

Consequently, set the significance level/confidence level before performing your analysis. Then, use those preset levels to determine statistical significance. I always recommend including the exact p-value when you report on statistical significance. Exact p-values do provide information about the strength of evidence against the null.

March 5, 2020 at 9:58 am

Thank you for sharing this knowledge , it is very appropriate in explaining some observations in the study of forest biodiversity.

March 4, 2020 at 2:01 am

Thank you so much. This provides for my research

March 3, 2020 at 7:28 pm

If one couples this with what they call estimated monetary value of risk in risk management, one can take better decisions.

March 3, 2020 at 3:12 pm

Thank you for providing this clear insight.

March 3, 2020 at 3:29 am

Nice article Jim. The risk of such failure obviously reduces when a lower significance level is specified.One benefits most by reading this article in conjunction with your other article “Understanding Significance Levels in Statistics”.

March 3, 2020 at 2:43 am

That’s fine. My question is why doesn’t the numerical value of type 1 error coincide with the significance level in the backdrop that the type 1 error and the significance level are both the same ? I hope you got my question.

March 3, 2020 at 3:30 am

Hi, they are equal. As I indicated, the significance level equals the type I error rate.

March 3, 2020 at 1:27 am

Kindly elighten me on one confusion. We set out our significance level before setting our hypothesis. When we calculate the type 1 error, which happens to be a significance level, the numerical value doesn’t equals (either undermining value comes out or an exceeding value comescout ) our significance level that was preassigned. Why is this so ?

March 3, 2020 at 2:24 am

Hi Ratnadeep,

You’re correct. The significance level (alpha) is the same as the type I error rate. However, you compare the p-value to the significance level. It’s the p-value that can be greater than or less than the significance level.

The significance level is the evidentiary standard. How strong does the evidence in your sample need to be before you can reject the null? The p-value indicates the strength of the evidence that is present in your sample. By comparing the p-value to the significance level, you’re comparing the actual strength of the sample evidence to the evidentiary standard to determine whether your sample evidence is strong enough to conclude that the effect exists in the population.

I write about this in my post about the understanding significance levels . I think that will help answer your questions!

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• When scientific hypotheses don’t pan out

One pair of scientists thought they’d discovered a new antiviral protein buried inside skin cells. Another research team saw early hints suggesting that the flu virus might cooperate to boost infections in humans. And a nationwide team of clinicians thought that high doses of certain vitamins might prevent cancer.

These studies don’t have much to do with each other, except that the researchers had all based their hypotheses on convincing earlier data.

And those hypotheses were all wrong.

The hypothesis is a central tenet to scientific research. Scientists ask questions, but a question on its own is often not sufficient to outline the experiments needed to answer it (nor to garner the funding needed to support those experiments).

So researchers construct a hypothesis, their best educated guess as to the answer to that question.

How a hypothesis is formed

Technically speaking, a hypothesis is only a hypothesis if it can be tested. Otherwise, it’s just an idea to discuss at the water cooler.

Researchers are always prepared for the possibility that those tests could disprove their hypotheses — that’s part of the reason they do the studies. But what happens when a beloved idea or dogma is shattered is less technical, less predictable. More human.

In some cases, a disproven hypothesis is devastating, said Swedish Cancer Institute and Fred Hutchinson Cancer Research Center public health researcher Dr. Gary Goodman, who led one of those vitamin studies. In his case, he was part of a group of cancer prevention researchers who ultimately showed that high doses of certain vitamins can increase the risk of lung cancer — an important result, but the opposite of what they thought they would prove in their trials.

But for some, finding a hypothesis to be false is exhilarating and motivating.

Herpes hypothesis leads to surprise cancer-related finding

Dr. Jia Zhu , a Fred Hutch infectious disease scientist, and her research partner (and husband), Fred Hutch and University of Washington infectious disease researcher Dr. Tao Peng, thought they’d found a new antiviral in herpes simplex virus type 2, or HSV-2, in part because they’ve been focused on that virus — and its interaction with human immune cells — for decades now, together with Dr. Larry Corey , virologist and president and director emeritus of Fred Hutch.

A few years ago, Zhu and Peng found that a tiny, mysterious protein called interleukin-17c is massively overproduced by HSV-infected skin cells. Maybe it was an undiscovered antiviral protein, the virologists thought, made by the skin cells in an attempt to protect themselves. They spent more than half a year pursuing that hypothesis, conducting experiment after experiment to see if IL-17c could block the herpes virus from replicating. It didn’t.

Zhu pointed to a microscopic image of a biopsy from a person with HSV, captured more than 10 years ago where she, Corey and their colleagues first discovered that certain T cells, a type of immune cell, cluster in the skin where herpes lesions form. At the top of the colorful image, a layer of skin cells stained blue is studded with orange-colored T cells. Beneath, green nerve endings stretch their branch-like fibers toward the infected skin cells.

“This is my favorite image, but we all focused on the top,” the skin and immune cells, Zhu said. “We never really paid attention to the nerves.”

"You take an approach and then you just have to let the science drive." — Dr. Jia Zhu, infectious disease researcher

Finally, Peng discovered that the nerve fibers themselves carry proteins that can interact with the IL-17c molecule produced in infected skin cells — and that the protein signals the nerves to grow, making it one of only a handful of nerve growth factors identified in humans.

The researchers are excited about their serendipitous finding not just because it’s another piece in the puzzle of this mysterious virus, which infects one in six teens and adults in the U.S. They also hope the protein could fuel new therapies in other settings — such as neuropathy, a type of nerve damage that is a side effect of many cancer chemotherapies.

It’s a finding they never would have anticipated, Zhu said, but that’s often the nature of research.

“You do have a big picture, you know the direction. You take an approach and then you just have to let the science drive,” she said. “If things are unexpected, maybe just explore a little bit more instead of shutting that door.”

Flu hypothesis leads to a new mindset and avenue of research

Sometimes, a mistaken hypothesis has less to do with researchers’ preconceptions and more to do with the way basic research is conducted. Take, for example, the work of Fred Hutch evolutionary biologist Dr. Jesse Bloom , whose laboratory team studies how influenza and other viruses evolve over time. Many of their experiments involve infecting human cells in a petri dish with different strains of the flu virus and seeing what happens.

A few years ago, Bloom and University of Washington doctoral student Katherine Xue made an intriguing discovery using that system: They saw that two variants of influenza H3N2 (the virus that’s wreaking havoc in the current flu season) could cooperate to infect cells better together than either version could alone.

The researchers had only shown that viral collaboration in petri dishes in the lab, but they had reason to think it might be happening in people, too. For one, the same mix of variants was present in public databases of samples taken from infected people — but those samples had also been grown in petri dishes in the lab before their genomic information was captured.

So Xue and Bloom sequenced those variants at their source, the original nasal wash samples collected and stored by the Washington State Public Health Laboratories . They found no such mixture of variants from the samples that hadn’t been grown in the laboratory — so the flu may not cooperate after all, at least not in our bodies. The researchers published their findings last month in the journal mSphere.

Scientists have to ask themselves two questions about any discovery, Bloom said: “Are your findings correct? And are they relevant?”

The team’s first study wasn’t wrong; the viruses do cooperate in cells in the lab. But the second question is usually the tougher one, the researchers said.

“There are a lot of differences, obviously, between viruses growing in a controlled setting in a petri dish versus an actual human,” Xue said.

She and Bloom aren’t too glum about their disproven hypothesis, though. That line of inquiry opened new doors in the lab, Bloom said.

Before Xue’s study, he and his colleagues exclusively studied viruses in petri dishes. Now, more members of his laboratory team are using clinical samples as well — an approach that is made possible by the closer collaborations between basic and clinical research at the Hutch, Bloom said.

Some of their findings in petri dishes aren’t holding true in the clinical samples. But they’re already making interesting findings about how flu evolves in the human body — including the discovery that how flu evolves in single people with unusually long infections can hint at how the virus will evolve globally, years later. They never would have done that study if they hadn’t already been trying to follow up their original, cooperating hypothesis.

“It opened this whole new way of trying to think about this,” Bloom said. “Our mindset has changed a lot.”

Prevention hypothesis flipped on its head

Fred Hutch and Swedish cancer prevention researcher Goodman and his epidemiology colleagues had good reason to think the vitamins they were testing in clinical trials could prevent lung cancer.

All of the data pointed to an association between the vitamins and a reduced risk of lung cancer. But the studies hadn’t shown a causative link — just a correlation. So the researchers set out to do large clinical trials comparing high doses of the vitamins to placebos.

In the CARET trial , which Goodman led and was initiated in 1985, 18,000 people at high risk of lung cancer (primarily smokers) were assigned to take either a placebo, vitamin A, beta-carotene (a vitamin A precursor) or a combination of the two supplements. Two other similar trials started in other parts of the world at around the same time also testing beta-carotene’s effect on lung cancer risk.

In a similar vein, at the same time, a small trial suggested that supplemental selenium decreased the incidence of prostate cancer. So in 2001, the SELECT trial launched through SWOG , a nationwide cancer clinical trial consortium, testing whether selenium or high-dose vitamin E or the combination could prevent prostate cancer. SELECT enrolled 35,000 men; Goodman was the study leader for the Seattle area.

Designing and conducting cancer prevention trials where participants take a drug or some other intervention is a tricky business, Goodman said.

“In prevention, most of the people you treat are healthy and will never get cancer,” he said. “So you have to make sure the agent is very safe.”

Previous studies had all pointed to the vitamins being safe — even beneficial. And the vitamins tested in the trials are all naturally occurring as part of our diets. Nobody thought they could possibly hurt.

But that’s exactly what happened. In the CARET study, participants taking the combination of vitamin A and beta-carotene had higher rates of lung cancer than those taking the placebo; other trials testing those vitamins saw similar results. And in the SELECT trial, those taking vitamin E had higher rates of prostate cancer.

All the trials had close monitoring built in and all were stopped early when the researchers saw that the cancer rates were trending the opposite way that they’d expected.

“It was just devastating when we learned the results,” Goodman said. “Everybody [who worked on the trial] was so hopeful. After all, we’re here to prevent cancer.”

When the CARET study stopped, Goodman and his team hired extra people to answer study participants’ questions and the angry phone calls they assumed they would get. But very few phone calls came in.

“They said they were involved in the study for altruistic reasons, and we got an answer,” he said. “One of the benefits of our study is that we did show that high doses of vitamins can be very harmful.”

That was an important finding, Goodman said, because the prevailing dogma at the time was that high doses of vitamins were good for you. Although these studies disproved that commonly held belief, even today not everyone in the general public buys that message.

Another benefit of that difficult experience: The bar for giving healthy people a supplement or drug with the goal of preventing cancer or other disease is much higher now, Goodman said.

“In prevention, [these studies] really changed people’s perceptions about what kind of evidence you need to have before you can invest the time, money, effort, human resources, people’s lives in an intervention study,” he said. “You really need to have good data suggesting that an intervention will be beneficial.”

Rachel Tompa is a former staff writer at Fred Hutchinson Cancer Center. She has a Ph.D. in molecular biology from the University of California, San Francisco and a certificate in science writing from the University of California, Santa Cruz. Follow her on Twitter @Rachel_Tompa .

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What is The Null Hypothesis & When Do You Reject The Null Hypothesis

Julia Simkus

Editor at Simply Psychology

BA (Hons) Psychology, Princeton University

Julia Simkus is a graduate of Princeton University with a Bachelor of Arts in Psychology. She is currently studying for a Master's Degree in Counseling for Mental Health and Wellness in September 2023. Julia's research has been published in peer reviewed journals.

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A null hypothesis is a statistical concept suggesting no significant difference or relationship between measured variables. It’s the default assumption unless empirical evidence proves otherwise.

The null hypothesis states no relationship exists between the two variables being studied (i.e., one variable does not affect the other).

The null hypothesis is the statement that a researcher or an investigator wants to disprove.

Testing the null hypothesis can tell you whether your results are due to the effects of manipulating ​ the dependent variable or due to random chance. 

How to Write a Null Hypothesis

Null hypotheses (H0) start as research questions that the investigator rephrases as statements indicating no effect or relationship between the independent and dependent variables.

It is a default position that your research aims to challenge or confirm.

For example, if studying the impact of exercise on weight loss, your null hypothesis might be:

There is no significant difference in weight loss between individuals who exercise daily and those who do not.

Examples of Null Hypotheses

When Do We Reject The Null Hypothesis? 

We reject the null hypothesis when the data provide strong enough evidence to conclude that it is likely incorrect. This often occurs when the p-value (probability of observing the data given the null hypothesis is true) is below a predetermined significance level.

If the collected data does not meet the expectation of the null hypothesis, a researcher can conclude that the data lacks sufficient evidence to back up the null hypothesis, and thus the null hypothesis is rejected. 

Rejecting the null hypothesis means that a relationship does exist between a set of variables and the effect is statistically significant ( p > 0.05).

If the data collected from the random sample is not statistically significance , then the null hypothesis will be accepted, and the researchers can conclude that there is no relationship between the variables. 

You need to perform a statistical test on your data in order to evaluate how consistent it is with the null hypothesis. A p-value is one statistical measurement used to validate a hypothesis against observed data.

Calculating the p-value is a critical part of null-hypothesis significance testing because it quantifies how strongly the sample data contradicts the null hypothesis.

The level of statistical significance is often expressed as a  p  -value between 0 and 1. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.

Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01) as general guidelines to decide if you should reject or keep the null.

When your p-value is less than or equal to your significance level, you reject the null hypothesis.

In other words, smaller p-values are taken as stronger evidence against the null hypothesis. Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis.

In this case, the sample data provides insufficient data to conclude that the effect exists in the population.

Because you can never know with complete certainty whether there is an effect in the population, your inferences about a population will sometimes be incorrect.

When you incorrectly reject the null hypothesis, it’s called a type I error. When you incorrectly fail to reject it, it’s called a type II error.

Why Do We Never Accept The Null Hypothesis?

The reason we do not say “accept the null” is because we are always assuming the null hypothesis is true and then conducting a study to see if there is evidence against it. And, even if we don’t find evidence against it, a null hypothesis is not accepted.

A lack of evidence only means that you haven’t proven that something exists. It does not prove that something doesn’t exist. 

It is risky to conclude that the null hypothesis is true merely because we did not find evidence to reject it. It is always possible that researchers elsewhere have disproved the null hypothesis, so we cannot accept it as true, but instead, we state that we failed to reject the null. 

One can either reject the null hypothesis, or fail to reject it, but can never accept it.

Why Do We Use The Null Hypothesis?

We can never prove with 100% certainty that a hypothesis is true; We can only collect evidence that supports a theory. However, testing a hypothesis can set the stage for rejecting or accepting this hypothesis within a certain confidence level.

The null hypothesis is useful because it can tell us whether the results of our study are due to random chance or the manipulation of a variable (with a certain level of confidence).

A null hypothesis is rejected if the measured data is significantly unlikely to have occurred and a null hypothesis is accepted if the observed outcome is consistent with the position held by the null hypothesis.

Rejecting the null hypothesis sets the stage for further experimentation to see if a relationship between two variables exists. 

Hypothesis testing is a critical part of the scientific method as it helps decide whether the results of a research study support a particular theory about a given population. Hypothesis testing is a systematic way of backing up researchers’ predictions with statistical analysis.

It helps provide sufficient statistical evidence that either favors or rejects a certain hypothesis about the population parameter. 

Purpose of a Null Hypothesis 

• The primary purpose of the null hypothesis is to disprove an assumption. 
• Whether rejected or accepted, the null hypothesis can help further progress a theory in many scientific cases.
• A null hypothesis can be used to ascertain how consistent the outcomes of multiple studies are.

Do you always need both a Null Hypothesis and an Alternative Hypothesis?

The null (H0) and alternative (Ha or H1) hypotheses are two competing claims that describe the effect of the independent variable on the dependent variable. They are mutually exclusive, which means that only one of the two hypotheses can be true. 

While the null hypothesis states that there is no effect in the population, an alternative hypothesis states that there is statistical significance between two variables. 

The goal of hypothesis testing is to make inferences about a population based on a sample. In order to undertake hypothesis testing, you must express your research hypothesis as a null and alternative hypothesis. Both hypotheses are required to cover every possible outcome of the study. 

What is the difference between a null hypothesis and an alternative hypothesis?

The alternative hypothesis is the complement to the null hypothesis. The null hypothesis states that there is no effect or no relationship between variables, while the alternative hypothesis claims that there is an effect or relationship in the population.

It is the claim that you expect or hope will be true. The null hypothesis and the alternative hypothesis are always mutually exclusive, meaning that only one can be true at a time.

What are some problems with the null hypothesis?

One major problem with the null hypothesis is that researchers typically will assume that accepting the null is a failure of the experiment. However, accepting or rejecting any hypothesis is a positive result. Even if the null is not refuted, the researchers will still learn something new.

Why can a null hypothesis not be accepted?

We can either reject or fail to reject a null hypothesis, but never accept it. If your test fails to detect an effect, this is not proof that the effect doesn’t exist. It just means that your sample did not have enough evidence to conclude that it exists.

We can’t accept a null hypothesis because a lack of evidence does not prove something that does not exist. Instead, we fail to reject it.

Failing to reject the null indicates that the sample did not provide sufficient enough evidence to conclude that an effect exists.

If the p-value is greater than the significance level, then you fail to reject the null hypothesis.

Is a null hypothesis directional or non-directional?

A hypothesis test can either contain an alternative directional hypothesis or a non-directional alternative hypothesis. A directional hypothesis is one that contains the less than (“<“) or greater than (“>”) sign.

A nondirectional hypothesis contains the not equal sign (“≠”).  However, a null hypothesis is neither directional nor non-directional.

A null hypothesis is a prediction that there will be no change, relationship, or difference between two variables.

The directional hypothesis or nondirectional hypothesis would then be considered alternative hypotheses to the null hypothesis.

Gill, J. (1999). The insignificance of null hypothesis significance testing.  Political research quarterly ,  52 (3), 647-674.

Krueger, J. (2001). Null hypothesis significance testing: On the survival of a flawed method.  American Psychologist ,  56 (1), 16.

Masson, M. E. (2011). A tutorial on a practical Bayesian alternative to null-hypothesis significance testing.  Behavior research methods ,  43 , 679-690.

Nickerson, R. S. (2000). Null hypothesis significance testing: a review of an old and continuing controversy.  Psychological methods ,  5 (2), 241.

Rozeboom, W. W. (1960). The fallacy of the null-hypothesis significance test.  Psychological bulletin ,  57 (5), 416.

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6a.1 - introduction to hypothesis testing, basic terms section  .

The first step in hypothesis testing is to set up two competing hypotheses. The hypotheses are the most important aspect. If the hypotheses are incorrect, your conclusion will also be incorrect.

The two hypotheses are named the null hypothesis and the alternative hypothesis.

The goal of hypothesis testing is to see if there is enough evidence against the null hypothesis. In other words, to see if there is enough evidence to reject the null hypothesis. If there is not enough evidence, then we fail to reject the null hypothesis.

Consider the following example where we set up these hypotheses.

Example 6-1 Section  

A man, Mr. Orangejuice, goes to trial and is tried for the murder of his ex-wife. He is either guilty or innocent. Set up the null and alternative hypotheses for this example.

Putting this in a hypothesis testing framework, the hypotheses being tested are:

• The man is guilty
• The man is innocent

Let's set up the null and alternative hypotheses.

\(H_0\colon \) Mr. Orangejuice is innocent

\(H_a\colon \) Mr. Orangejuice is guilty

Remember that we assume the null hypothesis is true and try to see if we have evidence against the null. Therefore, it makes sense in this example to assume the man is innocent and test to see if there is evidence that he is guilty.

The Logic of Hypothesis Testing Section  

We want to know the answer to a research question. We determine our null and alternative hypotheses. Now it is time to make a decision.

The decision is either going to be...

• reject the null hypothesis or...
• fail to reject the null hypothesis.

Consider the following table. The table shows the decision/conclusion of the hypothesis test and the unknown "reality", or truth. We do not know if the null is true or if it is false. If the null is false and we reject it, then we made the correct decision. If the null hypothesis is true and we fail to reject it, then we made the correct decision.

So what happens when we do not make the correct decision?

When doing hypothesis testing, two types of mistakes may be made and we call them Type I error and Type II error. If we reject the null hypothesis when it is true, then we made a type I error. If the null hypothesis is false and we failed to reject it, we made another error called a Type II error.

Types of errors

The “reality”, or truth, about the null hypothesis is unknown and therefore we do not know if we have made the correct decision or if we committed an error. We can, however, define the likelihood of these events.

\(\alpha\) and \(\beta\) are probabilities of committing an error so we want these values to be low. However, we cannot decrease both. As \(\alpha\) decreases, \(\beta\) increases.

Example 6-1 Cont'd... Section  

A man, Mr. Orangejuice, goes to trial and is tried for the murder of his ex-wife. He is either guilty or not guilty. We found before that...

• \( H_0\colon \) Mr. Orangejuice is innocent
• \( H_a\colon \) Mr. Orangejuice is guilty

Interpret Type I error, \(\alpha \), Type II error, \(\beta \).

As you can see here, the Type I error (putting an innocent man in jail) is the more serious error. Ethically, it is more serious to put an innocent man in jail than to let a guilty man go free. So to minimize the probability of a type I error we would choose a smaller significance level.

Try it! Section  

An inspector has to choose between certifying a building as safe or saying that the building is not safe. There are two hypotheses:

• Building is safe
• Building is not safe

Set up the null and alternative hypotheses. Interpret Type I and Type II error.

\( H_0\colon\) Building is not safe vs \(H_a\colon \) Building is safe

Power and \(\beta \) are complements of each other. Therefore, they have an inverse relationship, i.e. as one increases, the other decreases.

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Science News

Here’s why we care about attempts to prove the riemann hypothesis.

The latest effort shines a spotlight on an enduring prime numbers mystery

LINED UP   The Riemann zeta function has an infinite number of points where the function’s value is zero, located at the whirls of color in this plot. The Riemann hypothesis predicts that certain zeros lie along a single line, which is horizontal in this image, where the colorful bands meet the red.

Empetrisor/Wikimedia Commons ( CC BY-SA 4.0 )

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By Emily Conover

September 25, 2018 at 11:46 am

A famed mathematical enigma is once again in the spotlight.

The Riemann hypothesis, posited in 1859 by German mathematician Bernhard Riemann, is one of the biggest unsolved puzzles in mathematics. The hypothesis, which could unlock the mysteries of prime numbers, has never been proved. But mathematicians are buzzing about a new attempt.

Esteemed mathematician Michael Atiyah took a crack at proving the hypothesis in a lecture at the Heidelberg Laureate Forum in Germany on September 24. Despite the stature of Atiyah — who has won the two most prestigious honors in mathematics, the Fields Medal and the Abel Prize — many researchers have expressed skepticism about the proof. So the Riemann hypothesis remains up for grabs.

Let’s break down what the Riemann hypothesis is, and what a confirmed proof — if one is ever found — would mean for mathematics.

What is the Riemann hypothesis?

The Riemann hypothesis is a statement about a mathematical curiosity known as the Riemann zeta function. That function is closely entwined with prime numbers — whole numbers that are evenly divisible only by 1 and themselves. Prime numbers are mysterious: They are scattered in an inscrutable pattern across the number line, making it difficult to predict where each prime number will fall ( SN Online: 4/2/08 ).

But if the Riemann zeta function meets a certain condition, Riemann realized, it would reveal secrets of the prime numbers, such as how many primes exist below a given number. That required condition is the Riemann hypothesis. It conjectures that certain zeros of the function — the points where the function’s value equals zero — all lie along a particular line when plotted ( SN: 9/27/08, p. 14 ). If the hypothesis is confirmed, it could help expose a method to the primes’ madness.

Why is it so important?

Prime numbers are mathematical VIPs: Like atoms of the periodic table, they are the building blocks for larger numbers. Primes matter for practical purposes, too, as they are important for securing encrypted transmissions sent over the internet. And importantly, a multitude of mathematical papers take the Riemann hypothesis as a given. If this foundational assumption were proved correct, “many results that are believed to be true will be known to be true,” says mathematician Ken Ono of Emory University in Atlanta. “It’s a kind of mathematical oracle.”

Haven’t people tried to prove this before?

Yep. It’s difficult to count the number of attempts, but probably hundreds of researchers have tried their hands at a proof. So far none of the proofs have stood up to scrutiny. The problem is so stubborn that it now has a bounty on its head : The Clay Mathematics Institute has offered up $1 million to anyone who can prove the Riemann hypothesis.

Why is it so difficult to prove?

The Riemann zeta function is a difficult beast to work with. Even defining it is a challenge, Ono says. Furthermore, the function has an infinite number of zeros. If any one of those zeros is not on its expected line, the Riemann hypothesis is wrong. And since there are infinite zeros, manually checking each one won’t work. Instead, a proof must show without a doubt that no zero can be an outlier. For difficult mathematical quandaries like the Riemann hypothesis, the bar for acceptance of a proof is extremely high. Verification of such a proof typically requires months or even years of double-checking by other mathematicians before either everyone is convinced, or the proof is deemed flawed.

What will it take to prove the Riemann hypothesis?

Various mathematicians have made some amount of headway toward a proof. Ono likens it to attempting to climb Mount Everest and making it to base camp. While some clever mathematician may eventually be able to finish that climb, Ono says, “there is this belief that the ultimate proof … if one ever is made, will require a different level of mathematics.”

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An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors

Priya ranganathan.

1 Department of Anesthesiology, Critical Care and Pain, Tata Memorial Hospital, Mumbai, Maharashtra, India

2 Department of Surgical Oncology, Tata Memorial Centre, Mumbai, Maharashtra, India

The second article in this series on biostatistics covers the concepts of sample, population, research hypotheses and statistical errors.

How to cite this article

Ranganathan P, Pramesh CS. An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors. Indian J Crit Care Med 2019;23(Suppl 3):S230–S231.

Two papers quoted in this issue of the Indian Journal of Critical Care Medicine report. The results of studies aim to prove that a new intervention is better than (superior to) an existing treatment. In the ABLE study, the investigators wanted to show that transfusion of fresh red blood cells would be superior to standard-issue red cells in reducing 90-day mortality in ICU patients. 1 The PROPPR study was designed to prove that transfusion of a lower ratio of plasma and platelets to red cells would be superior to a higher ratio in decreasing 24-hour and 30-day mortality in critically ill patients. 2 These studies are known as superiority studies (as opposed to noninferiority or equivalence studies which will be discussed in a subsequent article).

SAMPLE VERSUS POPULATION

A sample represents a group of participants selected from the entire population. Since studies cannot be carried out on entire populations, researchers choose samples, which are representative of the population. This is similar to walking into a grocery store and examining a few grains of rice or wheat before purchasing an entire bag; we assume that the few grains that we select (the sample) are representative of the entire sack of grains (the population).

The results of the study are then extrapolated to generate inferences about the population. We do this using a process known as hypothesis testing. This means that the results of the study may not always be identical to the results we would expect to find in the population; i.e., there is the possibility that the study results may be erroneous.

HYPOTHESIS TESTING

A clinical trial begins with an assumption or belief, and then proceeds to either prove or disprove this assumption. In statistical terms, this belief or assumption is known as a hypothesis. Counterintuitively, what the researcher believes in (or is trying to prove) is called the “alternate” hypothesis, and the opposite is called the “null” hypothesis; every study has a null hypothesis and an alternate hypothesis. For superiority studies, the alternate hypothesis states that one treatment (usually the new or experimental treatment) is superior to the other; the null hypothesis states that there is no difference between the treatments (the treatments are equal). For example, in the ABLE study, we start by stating the null hypothesis—there is no difference in mortality between groups receiving fresh RBCs and standard-issue RBCs. We then state the alternate hypothesis—There is a difference between groups receiving fresh RBCs and standard-issue RBCs. It is important to note that we have stated that the groups are different, without specifying which group will be better than the other. This is known as a two-tailed hypothesis and it allows us to test for superiority on either side (using a two-sided test). This is because, when we start a study, we are not 100% certain that the new treatment can only be better than the standard treatment—it could be worse, and if it is so, the study should pick it up as well. One tailed hypothesis and one-sided statistical testing is done for non-inferiority studies, which will be discussed in a subsequent paper in this series.

STATISTICAL ERRORS

There are two possibilities to consider when interpreting the results of a superiority study. The first possibility is that there is truly no difference between the treatments but the study finds that they are different. This is called a Type-1 error or false-positive error or alpha error. This means falsely rejecting the null hypothesis.

The second possibility is that there is a difference between the treatments and the study does not pick up this difference. This is called a Type 2 error or false-negative error or beta error. This means falsely accepting the null hypothesis.

The power of the study is the ability to detect a difference between groups and is the converse of the beta error; i.e., power = 1-beta error. Alpha and beta errors are finalized when the protocol is written and form the basis for sample size calculation for the study. In an ideal world, we would not like any error in the results of our study; however, we would need to do the study in the entire population (infinite sample size) to be able to get a 0% alpha and beta error. These two errors enable us to do studies with realistic sample sizes, with the compromise that there is a small possibility that the results may not always reflect the truth. The basis for this will be discussed in a subsequent paper in this series dealing with sample size calculation.

Conventionally, type 1 or alpha error is set at 5%. This means, that at the end of the study, if there is a difference between groups, we want to be 95% certain that this is a true difference and allow only a 5% probability that this difference has occurred by chance (false positive). Type 2 or beta error is usually set between 10% and 20%; therefore, the power of the study is 90% or 80%. This means that if there is a difference between groups, we want to be 80% (or 90%) certain that the study will detect that difference. For example, in the ABLE study, sample size was calculated with a type 1 error of 5% (two-sided) and power of 90% (type 2 error of 10%) (1).

Table 1 gives a summary of the two types of statistical errors with an example

Statistical errors

In the next article in this series, we will look at the meaning and interpretation of ‘ p ’ value and confidence intervals for hypothesis testing.

Source of support: Nil

Conflict of interest: None

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Biology archive

Course: biology archive   >   unit 1, the scientific method.

• Controlled experiments
• The scientific method and experimental design

Introduction

• Make an observation.
• Ask a question.
• Form a hypothesis , or testable explanation.
• Make a prediction based on the hypothesis.
• Test the prediction.
• Iterate: use the results to make new hypotheses or predictions.

Scientific method example: Failure to toast

1. make an observation., 2. ask a question., 3. propose a hypothesis., 4. make predictions., 5. test the predictions..

• If the toaster does toast, then the hypothesis is supported—likely correct.
• If the toaster doesn't toast, then the hypothesis is not supported—likely wrong.

Logical possibility

Practical possibility, building a body of evidence, 6. iterate..

• If the hypothesis was supported, we might do additional tests to confirm it, or revise it to be more specific. For instance, we might investigate why the outlet is broken.
• If the hypothesis was not supported, we would come up with a new hypothesis. For instance, the next hypothesis might be that there's a broken wire in the toaster.

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Scientific Hypothesis, Model, Theory, and Law

Understanding the Difference Between Basic Scientific Terms

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Words have precise meanings in science. For example, "theory," "law," and "hypothesis" don't all mean the same thing. Outside of science, you might say something is "just a theory," meaning it's a supposition that may or may not be true. In science, however, a theory is an explanation that generally is accepted to be true. Here's a closer look at these important, commonly misused terms.

A hypothesis is an educated guess, based on observation. It's a prediction of cause and effect. Usually, a hypothesis can be supported or refuted through experimentation or more observation. A hypothesis can be disproven but not proven to be true.

Example: If you see no difference in the cleaning ability of various laundry detergents, you might hypothesize that cleaning effectiveness is not affected by which detergent you use. This hypothesis can be disproven if you observe a stain is removed by one detergent and not another. On the other hand, you cannot prove the hypothesis. Even if you never see a difference in the cleanliness of your clothes after trying 1,000 detergents, there might be one more you haven't tried that could be different.

Scientists often construct models to help explain complex concepts. These can be physical models like a model volcano or atom  or conceptual models like predictive weather algorithms. A model doesn't contain all the details of the real deal, but it should include observations known to be valid.

Example: The  Bohr model shows electrons orbiting the atomic nucleus, much the same way as the way planets revolve around the sun. In reality, the movement of electrons is complicated but the model makes it clear that protons and neutrons form a nucleus and electrons tend to move around outside the nucleus.

A scientific theory summarizes a hypothesis or group of hypotheses that have been supported with repeated testing. A theory is valid as long as there is no evidence to dispute it. Therefore, theories can be disproven. Basically, if evidence accumulates to support a hypothesis, then the hypothesis can become accepted as a good explanation of a phenomenon. One definition of a theory is to say that it's an accepted hypothesis.

Example: It is known that on June 30, 1908, in Tunguska, Siberia, there was an explosion equivalent to the detonation of about 15 million tons of TNT. Many hypotheses have been proposed for what caused the explosion. It was theorized that the explosion was caused by a natural extraterrestrial phenomenon , and was not caused by man. Is this theory a fact? No. The event is a recorded fact. Is this theory, generally accepted to be true, based on evidence to-date? Yes. Can this theory be shown to be false and be discarded? Yes.

A scientific law generalizes a body of observations. At the time it's made, no exceptions have been found to a law. Scientific laws explain things but they do not describe them. One way to tell a law and a theory apart is to ask if the description gives you the means to explain "why." The word "law" is used less and less in science, as many laws are only true under limited circumstances.

Example: Consider Newton's Law of Gravity . Newton could use this law to predict the behavior of a dropped object but he couldn't explain why it happened.

As you can see, there is no "proof" or absolute "truth" in science. The closest we get are facts, which are indisputable observations. Note, however, if you define proof as arriving at a logical conclusion, based on the evidence, then there is "proof" in science. Some work under the definition that to prove something implies it can never be wrong, which is different. If you're asked to define the terms hypothesis, theory, and law, keep in mind the definitions of proof and of these words can vary slightly depending on the scientific discipline. What's important is to realize they don't all mean the same thing and cannot be used interchangeably.

• Theory Definition in Science
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• The Continental Drift Theory: Revolutionary and Significant
• Hypothesis Definition (Science)
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• Hypothesis, Model, Theory, and Law
• Science Projects for Every Subject
• Is Anthropology a Science?
• A Brief History of Atomic Theory
• Usage and Examples of a Rebuttal
• Fallacies of Relevance: Appeal to Authority
• Social Constructionism Definition and Examples
• Scientific Hypothesis Examples
• What Is Belief Perseverance? Definition and Examples

Forget what you’ve read, science can’t prove a thing

Senior Lecturer in the School of Physics, University of Sydney

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Dr. Biercuk conducts experimental research in quantum physics and quantum control. He is funded by the ARC Centre for Engineered Quantum Systems and the US Army Research Office. His work is not connected to climate science.

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Do scientists have a language problem? Do policy makers have hearing issues?

It would certainly seem so. Of late there have been frequent lamentations about scientists’ failure to make their case to the public on hot-button issues, or of policy makers to listen to their input.

Often scientists and the public are, in fact, communicating, but they’re talking right past one another. So what’s going on?

Words, words, words

It’s imperative scientists do a better job communicating the meaning of the words they choose.

I don’t mean jargon: I mean basic words that mean one thing in scientific circles and another to the wider public.

Let’s take the climate change debate, and focus on a few specific words in context.

1) Theory : “Climate change is just a theory ”

In science effectively all ideas are “just” theories.

Scientists often use concepts from the philosophy of science to make some semantic distinctions between laws, theories, hypotheses, and the like.

So when a scientist talks about a “law” of nature, he or she is referring only to a standard observation (given some strict parameters), not an absolute requirement.

A basic principle in science is that any law, theory, or otherwise can be disproven if new facts or evidence are presented.

If it cannot be somehow disproven by an experiment, then it is not scientific.

Take, for example, the Universal Law of Gravitation .

This “law” describes the motion of heavenly bodies, and how we stay firmly planted on the ground.

But this “law” is in fact not always right – it just captures what we usually observe.

In this case, we are ultimately referring to the “theory” of gravity – a theory supported by a huge body of evidence, but still just a theory.

The same is the case for human-induced climate change.

2) Proof : “We can’t prove humans are causing climate change.”

When people ask for proof, they generally just mean “evidence”. Scientists may have lots of “evidence”, but will never claim to have “proof,” because proof does not exist in science .

Proof has a technical meaning that only applies in mathematics.

All we can do in science is collect evidence – lots of it – much the way we do in testing gravitational theory.

So long as the evidence is consistent with the theory, we consider the theory validated. But it will never be proven.

A critic or sceptic may view a scientist’s hedging on the issue of “proof” as a sign of weakness – really it’s just a sign the scientist’s meaning of the word is different to the general public’s.

3) Uncertainty : “Even proponents of human-induced climate change admit uncertainty .”

This use of “uncertainty” is perhaps the stickiest issue. In the public mind, uncertainty suggests someone is confused or otherwise not fully in command of the material.

But in science, uncertainty is always present…because nothing can be proven.

Is this a bad thing? No.

A scientist cannot and will not say there is absolute certainty that you won’t fly off into space due to a quirk of gravitational theory, quantum mechanics, or the like.

He or she would simply say that such an occurrence is extraordinarily unlikely – so unlikely that it wouldn’t, on average, happen even once during the age of the known universe. Even though that’s really unlikely there remains uncertainty.

In the climate change debate the uncertainty is much larger than in this example, but it is fundamentally no different to the scientist’s mind.

A winning formula

The burden for fixing this communication problem falls most heavily on the scientist. We need to better educate the public about the meaning of our words, and of the basic principles of science. We fail when we assume that everyone thinks or argues the way that we do.

This is a huge challenge in an age of 24-7 news channels that are looking for three-word soundbites.

No-one wants to diminish the accuracy of our statements. But even accommodating the modern media cycle is possible.

So how about this?

In science …

Everything’s a theory.

Proof doesn’t exist.

Nothing is certain.

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• Public awareness of science

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Methodology

• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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McCombes, S. (2023, November 20). How to Write a Strong Hypothesis | Steps & Examples. Scribbr. Retrieved July 19, 2024, from https://www.scribbr.com/methodology/hypothesis/

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Responding to a Disproven or Failed Research Hypothesis

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• 03 August, 2022

When meeting with a disproven or failed hypothesis , after having expended so much time and effort, precisely how should researchers respond? Responding well to a disproven or failed hypothesis is an essential component to scientific research . As a researcher, it helps to learn ‘ research resilience ’: the ability to carefully analyse, effectively document and broadly disseminate the failed hypotheses, all with an eye towards learning and future progress. This article explores common reasons why a hypothesis fails, as well as specific ways you can respond and lessons you can learn from this. 

Note : This article assumes that you are working on a hypothesis (not a null hypothesis): in other words, you are seeking to prove that the hypothesis is true, rather than to disprove it. 

Reasons why a hypothesis is disproven/fails

Hypotheses are disproved or fail for a number of reasons, including:

• The researcher’s preconception is incorrect , which leads to a flawed and failed hypothesis.
• The researcher’s findings are correct, but those findings aren’t relevant .
• Data set/sample size may not be sufficiently large to yield meaningful results. (If interested, learn more about this here: The importance of having Large Sample Sizes for your research )
• The hypothesis itself lies outside the realm of science . The hypothesis cannot be tested by experiments for which results have the potential to show that the idea is false.

Responding to a disproved hypothesis

After weeks or even months of intense thinking and experimenting, you have come to the conclusion that your hypothesis is disproven. So, what can you do to respond to such a disheartening realisation? Here are some practical steps you can take.

• Analyse the hypothesis carefully, as well as your research.   Performing a rigorous, methodical ‘post-mortem’ evaluation of your hypothesis and experiments will enable you to learn from them and to effectively and efficiently share your reflections with others. Use the following questions to evaluate how the research was conducted: 
• Did you conduct the experiment(s) correctly? 
• Was the study sufficiently powered to truly provide a definitive answer?
• Would a larger, better powered study – possibly conducted collaboratively with other research centres – be necessary, appropriate or helpful? 
• Would altering the experiment — or conducting different experiments — more appropriately answer your hypothesis? 
• Share the disproven hypothesis, and your experiments and analysis, with colleagues. Sharing negative data can help to interpret positive results from related studies and can aid you to adjust your experimental design .
• Consider the possibility that the hypothesis was not an attempt at gaining true scientific understanding, but rather, was a measure of a prevailing bias .

Positive lessons to be gained from a disproved hypothesis

Even the most successful, creative and thoughtful researchers encounter failed hypotheses. What makes them stand out is their ability to learn from failure. The following considerations may assist you to learn and gain from failed hypotheses:

• Failure can be beneficial if it leads directly toward future exploration.
• Does the failed hypothesis definitively close the door on further research? If so, such definitive knowledge is progress.
• Does the failed hypothesis simply point to the need to wait for a future date when more refined experiments or analysis can be conducted? That knowledge, too, is useful. 
• ‘Atomising’ (breaking down and dissecting) the reasoning behind the conceptual foundation of the failed hypothesis may uncover flawed yet correctable thinking in how the hypothesis was developed. 
• Failure leads to investigation and creativity in the pursuit of viable alternative hypotheses, experiments and statistical analyses. Better theoretical or experimental models often arise out of the ashes of a failed hypothesis, as do studies with more rigorously attained evidence (such as larger-scale, low-bias meta-analyses ). 

Considering a post-hoc analysis

A failed hypothesis can then prompt you to conduct a post-hoc analysis. (If interested, learn more about it here: Significance and use of Post-hoc Analysis studies )

All is not lost if you conclude you have a failed hypothesis. Remember: A hypothesis can’t be right unless it can be proven wrong.  Developing research resilience will reward you with long-term success.

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Why not just testing alternative hypothesis? Why do we need null hypothesis?

For example, I am testing the effectiveness of a new drug. I can choose two groups: control and experimental. Based on the result I can say that whether the drug is working or not. Why do I need to state the null hypothesis in this case anyway?

• hypothesis-testing
• descriptive-statistics

3 Answers 3

Welcome to Cross Validated!

I'm sure someone could give a more canonical answer, but here's the conceptual gist of it.

Think of it this way: there is only one null hypothesis, right? The hypothesis that there is no difference between your two samples/populations.

However, how could you define your alternative hypothesis? There could be infinite alternative hypotheses. If you have a statistically significant difference between your control and experimental group, that could be due to:

• Demographic (age, sex, ethnicity etc.) differences between your groups
• Difference in socio-economic status of your two groups
• Difference in diet between your two groups
• Difference in unknown underlying genetic factors
• Difference in epigenetic factors between the two groups, due to varied prenatal/childhood experiences
• Difference everyday behavior (so the placebo didn't work, or worked "extra well")
• The drug actually worked

I am not a researcher, and I don't know your experiment, so these are just guesses. But I hope you can see the point - there is only one null hypothesis (nothing is different between the groups) vs. an infinite number of possible explanations for an observed difference between the two groups. You assume that it's your intervention/drug that is the difference, but it could be anything! This is the conceptual reason why you don't test an alternative hypothesis - because which one would you test? What would be your assumed effect size? Too many (literally infinite) possibilities and assumptions. In vast majority of cases, the only thing you can actually test is "Is there some difference between these two groups?", and you set up the experiment as best you can so that if there is a difference, you can attribute it to the intervention (because you controlled for demographics, diet, behavioral factors etc.)

I hope this helped clarify your understanding!

In a test of hypothesis, the null hypothesis provides the distribution of the test statistic, and thus the critical value or the P-value, either of which can be used to decide whether to reject.

Three elementary examples:

For normal data with known $\sigma,$ we can test $H_0: \mu = 5$ against $H_a: \mu \ne 5.$ Then the test statistic is $Z = \frac{\bar X - 5}{\sigma/\sqrt{n}} \sim \mathsf{Norm}(0,1),$ so that we reject $H_0$ at the 5% level, if $|Z| \ge 1.96.$

In a chi-squared test of independence for a table of observed counts $X_{ij},$ the null hypothesis provides a formula for finding the expected counts $E_{ij}$ to evaluate the test statistic $Q = \sum_{ij} \frac{(X_{ij}-E_{ij})^2}{E_{ij}}$ distributed approximately as $\mathsf{Chisq}(\nu),$ where $\nu = (r-1)(c-1),$ if the table has $r$ rows and $c$ columns.

Suppose we have $X = 9$ successes in $n = 25$ Bernoulli trials with Success probability $p$ and want to test $H_0: p = 0.5$ against $H_a: p < 0.5$ Then the test statistic for an exact binomial test is $X \sim \mathsf{Binom}(25, 0.5).$ In R, the procedure binom.test is shown below, where the P-value, not leading to rejection, is computed using this binomial distribution.

because alternative hypothesis (Ha) will have a certain degree of correlation attached to variables, like strongly correlated or weakly correlated. Suppose you simply state Ha, then ppl will ask how strong is the correlation between variables. Suppose its weak correlation, again ppl will question, how much weaker? suppose Ha is very weak almost close to Zero, in such cases you'll need Null hypothesis (Ho) to make a distinction between Ha and Ho

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Hypothesis Testing – A Deep Dive into Hypothesis Testing, The Backbone of Statistical Inference

• September 21, 2023

Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions.

In this Blog post we will learn:

• What is Hypothesis Testing?
• Steps in Hypothesis Testing 2.1. Set up Hypotheses: Null and Alternative 2.2. Choose a Significance Level (α) 2.3. Calculate a test statistic and P-Value 2.4. Make a Decision
• Example : Testing a new drug.
• Example in python

1. What is Hypothesis Testing?

In simple terms, hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. Imagine being handed a dice and asked if it’s biased. By rolling it a few times and analyzing the outcomes, you’d be engaging in the essence of hypothesis testing.

Think of hypothesis testing as the scientific method of the statistics world. Suppose you hear claims like “This new drug works wonders!” or “Our new website design boosts sales.” How do you know if these statements hold water? Enter hypothesis testing.

2. Steps in Hypothesis Testing

• Set up Hypotheses : Begin with a null hypothesis (H0) and an alternative hypothesis (Ha).
• Choose a Significance Level (α) : Typically 0.05, this is the probability of rejecting the null hypothesis when it’s actually true. Think of it as the chance of accusing an innocent person.
• Calculate Test statistic and P-Value : Gather evidence (data) and calculate a test statistic.
• p-value : This is the probability of observing the data, given that the null hypothesis is true. A small p-value (typically ≤ 0.05) suggests the data is inconsistent with the null hypothesis.
• Decision Rule : If the p-value is less than or equal to α, you reject the null hypothesis in favor of the alternative.

2.1. Set up Hypotheses: Null and Alternative

Before diving into testing, we must formulate hypotheses. The null hypothesis (H0) represents the default assumption, while the alternative hypothesis (H1) challenges it.

For instance, in drug testing, H0 : “The new drug is no better than the existing one,” H1 : “The new drug is superior .”

2.2. Choose a Significance Level (α)

When You collect and analyze data to test H0 and H1 hypotheses. Based on your analysis, you decide whether to reject the null hypothesis in favor of the alternative, or fail to reject / Accept the null hypothesis.

The significance level, often denoted by $α$, represents the probability of rejecting the null hypothesis when it is actually true.

In other words, it’s the risk you’re willing to take of making a Type I error (false positive).

Type I Error (False Positive) :

• Symbolized by the Greek letter alpha (α).
• Occurs when you incorrectly reject a true null hypothesis . In other words, you conclude that there is an effect or difference when, in reality, there isn’t.
• The probability of making a Type I error is denoted by the significance level of a test. Commonly, tests are conducted at the 0.05 significance level , which means there’s a 5% chance of making a Type I error .
• Commonly used significance levels are 0.01, 0.05, and 0.10, but the choice depends on the context of the study and the level of risk one is willing to accept.

Example : If a drug is not effective (truth), but a clinical trial incorrectly concludes that it is effective (based on the sample data), then a Type I error has occurred.

Type II Error (False Negative) :

• Symbolized by the Greek letter beta (β).
• Occurs when you accept a false null hypothesis . This means you conclude there is no effect or difference when, in reality, there is.
• The probability of making a Type II error is denoted by β. The power of a test (1 – β) represents the probability of correctly rejecting a false null hypothesis.

Example : If a drug is effective (truth), but a clinical trial incorrectly concludes that it is not effective (based on the sample data), then a Type II error has occurred.

Balancing the Errors :

In practice, there’s a trade-off between Type I and Type II errors. Reducing the risk of one typically increases the risk of the other. For example, if you want to decrease the probability of a Type I error (by setting a lower significance level), you might increase the probability of a Type II error unless you compensate by collecting more data or making other adjustments.

It’s essential to understand the consequences of both types of errors in any given context. In some situations, a Type I error might be more severe, while in others, a Type II error might be of greater concern. This understanding guides researchers in designing their experiments and choosing appropriate significance levels.

2.3. Calculate a test statistic and P-Value

Test statistic : A test statistic is a single number that helps us understand how far our sample data is from what we’d expect under a null hypothesis (a basic assumption we’re trying to test against). Generally, the larger the test statistic, the more evidence we have against our null hypothesis. It helps us decide whether the differences we observe in our data are due to random chance or if there’s an actual effect.

P-value : The P-value tells us how likely we would get our observed results (or something more extreme) if the null hypothesis were true. It’s a value between 0 and 1. – A smaller P-value (typically below 0.05) means that the observation is rare under the null hypothesis, so we might reject the null hypothesis. – A larger P-value suggests that what we observed could easily happen by random chance, so we might not reject the null hypothesis.

2.4. Make a Decision

Relationship between $α$ and P-Value

When conducting a hypothesis test:

We then calculate the p-value from our sample data and the test statistic.

Finally, we compare the p-value to our chosen $α$:

• If $p−value≤α$: We reject the null hypothesis in favor of the alternative hypothesis. The result is said to be statistically significant.
• If $p−value>α$: We fail to reject the null hypothesis. There isn’t enough statistical evidence to support the alternative hypothesis.

3. Example : Testing a new drug.

Imagine we are investigating whether a new drug is effective at treating headaches faster than drug B.

Setting Up the Experiment : You gather 100 people who suffer from headaches. Half of them (50 people) are given the new drug (let’s call this the ‘Drug Group’), and the other half are given a sugar pill, which doesn’t contain any medication.

• Set up Hypotheses : Before starting, you make a prediction:
• Null Hypothesis (H0): The new drug has no effect. Any difference in healing time between the two groups is just due to random chance.
• Alternative Hypothesis (H1): The new drug does have an effect. The difference in healing time between the two groups is significant and not just by chance.

Calculate Test statistic and P-Value : After the experiment, you analyze the data. The “test statistic” is a number that helps you understand the difference between the two groups in terms of standard units.

For instance, let’s say:

• The average healing time in the Drug Group is 2 hours.
• The average healing time in the Placebo Group is 3 hours.

The test statistic helps you understand how significant this 1-hour difference is. If the groups are large and the spread of healing times in each group is small, then this difference might be significant. But if there’s a huge variation in healing times, the 1-hour difference might not be so special.

Imagine the P-value as answering this question: “If the new drug had NO real effect, what’s the probability that I’d see a difference as extreme (or more extreme) as the one I found, just by random chance?”

For instance:

• P-value of 0.01 means there’s a 1% chance that the observed difference (or a more extreme difference) would occur if the drug had no effect. That’s pretty rare, so we might consider the drug effective.
• P-value of 0.5 means there’s a 50% chance you’d see this difference just by chance. That’s pretty high, so we might not be convinced the drug is doing much.
• If the P-value is less than ($α$) 0.05: the results are “statistically significant,” and they might reject the null hypothesis , believing the new drug has an effect.
• If the P-value is greater than ($α$) 0.05: the results are not statistically significant, and they don’t reject the null hypothesis , remaining unsure if the drug has a genuine effect.

4. Example in python

For simplicity, let’s say we’re using a t-test (common for comparing means). Let’s dive into Python:

Making a Decision : “The results are statistically significant! p-value < 0.05 , The drug seems to have an effect!” If not, we’d say, “Looks like the drug isn’t as miraculous as we thought.”

5. Conclusion

Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.

More Articles

Correlation – connecting the dots, the role of correlation in data analysis, sampling and sampling distributions – a comprehensive guide on sampling and sampling distributions, law of large numbers – a deep dive into the world of statistics, central limit theorem – a deep dive into central limit theorem and its significance in statistics, skewness and kurtosis – peaks and tails, understanding data through skewness and kurtosis”, similar articles, complete introduction to linear regression in r, how to implement common statistical significance tests and find the p value, logistic regression – a complete tutorial with examples in r.

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Is it possible not to have a hypothesis in your thesis? [closed]

I have been working on a research report in which forecasting is done on the basis of data, followed by an interpretation of the forecasted results.

Is it possible to have that kind of research without hypothesizing any statement?

If this question is off-topic kindly recommend a suitable community.

• research-process
• methodology

• Would you please provide context for your thesis. For example, what is your area of study and what subfield are you in? –  Richard Erickson Commented Jun 14, 2017 at 17:50
• It would be answerable, if you could please add some more details as suggsted by @RichardErickson. –  Coder Commented Jun 14, 2017 at 18:02
• 3 This question is not about academia but about statistics. It might be on-topic on crossvalidated.se –  henning no longer feeds AI Commented Jun 15, 2017 at 5:58
• 2 @henning I made it about statistics, but the question as it stands is about research protocol. On crossvalidated Ibn e Ashiq can ask how to do the statistics. –  Joris Meys Commented Jun 15, 2017 at 8:52

As a statistician, I'm inclined to say "no, you can't" as a short answer. Reason for this is simple: even in complete random datasets on average 5% of the correlations will be significant when tested in a model. So if you rely on only the data to make any kind of interpretation on association of variables, you're bound to publish false positives. This has been discussed for decades, eg this rather strong opinion of Ioannidis (2005) :

http://journals.plos.org/plosmedicine/article?id=10.1371/journal.pmed.0020124

Bias can entail manipulation in the analysis or reporting of findings. Selective or distorted reporting is a typical form of such bias.

If you don't formulate a hypothesis and still interprete the data, you're -probably unintentionally- reporting selectively. You select from the analysis those results that tell a story, and in doing so, you're likely to report something that isn't a solid association or relation.

That said, you don't always have to formulate a specific hypothesis. For example, if you compare multiple methods on efficiency, you don't have to hypothesize beforehand which one is going to be the best. But the statistical test you use for comparison, will imply a "null hypothesis" that there is no real difference between all methods. Also this is "formulating a hypothesis" merely by the choice of analysis tools.

And this is even more important to realize: you might not formulate a hypothesis explicitly, but the nature of the statistical tools you use to come to your interpretation, will imply a set of rather rigid hypotheses and assumptions. You need to be aware of those hypotheses and also of those assumptions.

Because that's something I see far too often: people not explicitly formulating a hypothesis, still interpreting results from a statistical methodology, but failing to realize that their data does not meet the assumptions of that methodology. And that invalidates your entire interpretation.

This problem is even more stringent when forecasting. If you use regression models, you should be aware that predictions outside the boundaries of the original data cannot be interpreted. The uncertainty on those predictions is simply too big. If you use spline methods, you can even get into trouble at the edge of your original data. So definitely in the case of forecasting I would write out both the goal of the research and what you expect the predictions to show, including the scientific reason why. Only in those cases you can use forecasts as some form of evidence for or against the expected relation. If you don't do that, your forecasting model might as well be a fancy random number generator.

So in conclusion: even if your research goal isn't necessarily a defined hypothesis, you still need to formulate the hypotheses you want to test before carrying out the actual statistical tests.

And in all honesty, writing down what you expect to see is always a good idea, even if it's only to order your own thoughts.

• 1 +1 This is a really important point and a core principle of the scientific method, but for some reason many people either ignore it (out of convenience) or do not know about it. –  101010111100 Commented Jun 14, 2017 at 18:38
• There are fields - especially in Engineering - where results do not depend on a statistical analysis so as non-statistician, I would say, yes you can write a thesis without a hypothesis. –  o4tlulz Commented Jun 15, 2017 at 4:04
• 1 @Kevin That's using cross-validation, an often used statistical technique with its own assumptions. If you do that, you have to keep in mind that your training data and testing data have to be completely independent, or any hypothesis testing (yes, also there you test a hypothesis) is invalid. Independence is one of the most important assumptions in about every common statistical technique. –  Joris Meys Commented Jun 15, 2017 at 8:47
• 1 @Ooker I always tell my students to only use methods they know and understand. You can't know everything about statistics. But what you need to know, is all the details of the techniques you use yourself. And in any case every student should have the knowledge of the basic tests used in the majority of papers. Because if you don't understand those, there's no way you can evaluate yourself whether the conclusion in a paper actually makes sense. –  Joris Meys Commented Jun 15, 2017 at 8:51
• 1 @o4tlulz and how do you assure me that it solves the issue? Can you prove that? Can you show me your "solution" isn't just random luck? –  Joris Meys Commented Jun 16, 2017 at 7:45

Not the answer you're looking for? Browse other questions tagged phd research-process thesis methodology .

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What Is a Hypothesis and How Do I Write One?

General Education

Think about something strange and unexplainable in your life. Maybe you get a headache right before it rains, or maybe you think your favorite sports team wins when you wear a certain color. If you wanted to see whether these are just coincidences or scientific fact, you would form a hypothesis, then create an experiment to see whether that hypothesis is true or not.

But what is a hypothesis, anyway? If you’re not sure about what a hypothesis is--or how to test for one!--you’re in the right place. This article will teach you everything you need to know about hypotheses, including: 

• Defining the term “hypothesis” 
• Providing hypothesis examples 
• Giving you tips for how to write your own hypothesis

So let’s get started!

What Is a Hypothesis?

Merriam Webster defines a hypothesis as “an assumption or concession made for the sake of argument.” In other words, a hypothesis is an educated guess . Scientists make a reasonable assumption--or a hypothesis--then design an experiment to test whether it’s true or not. Keep in mind that in science, a hypothesis should be testable. You have to be able to design an experiment that tests your hypothesis in order for it to be valid. 

As you could assume from that statement, it’s easy to make a bad hypothesis. But when you’re holding an experiment, it’s even more important that your guesses be good...after all, you’re spending time (and maybe money!) to figure out more about your observation. That’s why we refer to a hypothesis as an educated guess--good hypotheses are based on existing data and research to make them as sound as possible.

Hypotheses are one part of what’s called the scientific method .  Every (good) experiment or study is based in the scientific method. The scientific method gives order and structure to experiments and ensures that interference from scientists or outside influences does not skew the results. It’s important that you understand the concepts of the scientific method before holding your own experiment. Though it may vary among scientists, the scientific method is generally made up of six steps (in order):

• Observation
• Asking questions
• Forming a hypothesis
• Analyze the data
• Communicate your results

You’ll notice that the hypothesis comes pretty early on when conducting an experiment. That’s because experiments work best when they’re trying to answer one specific question. And you can’t conduct an experiment until you know what you’re trying to prove!

Independent and Dependent Variables 

After doing your research, you’re ready for another important step in forming your hypothesis: identifying variables. Variables are basically any factor that could influence the outcome of your experiment . Variables have to be measurable and related to the topic being studied.

There are two types of variables:  independent variables and dependent variables. I ndependent variables remain constant . For example, age is an independent variable; it will stay the same, and researchers can look at different ages to see if it has an effect on the dependent variable. 

Speaking of dependent variables... dependent variables are subject to the influence of the independent variable , meaning that they are not constant. Let’s say you want to test whether a person’s age affects how much sleep they need. In that case, the independent variable is age (like we mentioned above), and the dependent variable is how much sleep a person gets. 

Variables will be crucial in writing your hypothesis. You need to be able to identify which variable is which, as both the independent and dependent variables will be written into your hypothesis. For instance, in a study about exercise, the independent variable might be the speed at which the respondents walk for thirty minutes, and the dependent variable would be their heart rate. In your study and in your hypothesis, you’re trying to understand the relationship between the two variables.

Elements of a Good Hypothesis

The best hypotheses start by asking the right questions . For instance, if you’ve observed that the grass is greener when it rains twice a week, you could ask what kind of grass it is, what elevation it’s at, and if the grass across the street responds to rain in the same way. Any of these questions could become the backbone of experiments to test why the grass gets greener when it rains fairly frequently.

As you’re asking more questions about your first observation, make sure you’re also making more observations . If it doesn’t rain for two weeks and the grass still looks green, that’s an important observation that could influence your hypothesis. You'll continue observing all throughout your experiment, but until the hypothesis is finalized, every observation should be noted.

Finally, you should consult secondary research before writing your hypothesis . Secondary research is comprised of results found and published by other people. You can usually find this information online or at your library. Additionally, m ake sure the research you find is credible and related to your topic. If you’re studying the correlation between rain and grass growth, it would help you to research rain patterns over the past twenty years for your county, published by a local agricultural association. You should also research the types of grass common in your area, the type of grass in your lawn, and whether anyone else has conducted experiments about your hypothesis. Also be sure you’re checking the quality of your research . Research done by a middle school student about what minerals can be found in rainwater would be less useful than an article published by a local university.

Writing Your Hypothesis

Once you’ve considered all of the factors above, you’re ready to start writing your hypothesis. Hypotheses usually take a certain form when they’re written out in a research report.

When you boil down your hypothesis statement, you are writing down your best guess and not the question at hand . This means that your statement should be written as if it is fact already, even though you are simply testing it.

The reason for this is that, after you have completed your study, you'll either accept or reject your if-then or your null hypothesis. All hypothesis testing examples should be measurable and able to be confirmed or denied. You cannot confirm a question, only a statement! 

In fact, you come up with hypothesis examples all the time! For instance, when you guess on the outcome of a basketball game, you don’t say, “Will the Miami Heat beat the Boston Celtics?” but instead, “I think the Miami Heat will beat the Boston Celtics.” You state it as if it is already true, even if it turns out you’re wrong. You do the same thing when writing your hypothesis.

Additionally, keep in mind that hypotheses can range from very specific to very broad.  These hypotheses can be specific, but if your hypothesis testing examples involve a broad range of causes and effects, your hypothesis can also be broad.  

The Two Types of Hypotheses

Now that you understand what goes into a hypothesis, it’s time to look more closely at the two most common types of hypothesis: the if-then hypothesis and the null hypothesis.

#1: If-Then Hypotheses

First of all, if-then hypotheses typically follow this formula:

If ____ happens, then ____ will happen.

The goal of this type of hypothesis is to test the causal relationship between the independent and dependent variable. It’s fairly simple, and each hypothesis can vary in how detailed it can be. We create if-then hypotheses all the time with our daily predictions. Here are some examples of hypotheses that use an if-then structure from daily life: 

• If I get enough sleep, I’ll be able to get more work done tomorrow.
• If the bus is on time, I can make it to my friend’s birthday party. 
• If I study every night this week, I’ll get a better grade on my exam. 

In each of these situations, you’re making a guess on how an independent variable (sleep, time, or studying) will affect a dependent variable (the amount of work you can do, making it to a party on time, or getting better grades). 

You may still be asking, “What is an example of a hypothesis used in scientific research?” Take one of the hypothesis examples from a real-world study on whether using technology before bed affects children’s sleep patterns. The hypothesis read s:

“We hypothesized that increased hours of tablet- and phone-based screen time at bedtime would be inversely correlated with sleep quality and child attention.”

It might not look like it, but this is an if-then statement. The researchers basically said, “If children have more screen usage at bedtime, then their quality of sleep and attention will be worse.” The sleep quality and attention are the dependent variables and the screen usage is the independent variable. (Usually, the independent variable comes after the “if” and the dependent variable comes after the “then,” as it is the independent variable that affects the dependent variable.) This is an excellent example of how flexible hypothesis statements can be, as long as the general idea of “if-then” and the independent and dependent variables are present.

#2: Null Hypotheses

Your if-then hypothesis is not the only one needed to complete a successful experiment, however. You also need a null hypothesis to test it against. In its most basic form, the null hypothesis is the opposite of your if-then hypothesis . When you write your null hypothesis, you are writing a hypothesis that suggests that your guess is not true, and that the independent and dependent variables have no relationship .

One null hypothesis for the cell phone and sleep study from the last section might say: 

“If children have more screen usage at bedtime, their quality of sleep and attention will not be worse.” 

In this case, this is a null hypothesis because it’s asking the opposite of the original thesis! 

Conversely, if your if-then hypothesis suggests that your two variables have no relationship, then your null hypothesis would suggest that there is one. So, pretend that there is a study that is asking the question, “Does the amount of followers on Instagram influence how long people spend on the app?” The independent variable is the amount of followers, and the dependent variable is the time spent. But if you, as the researcher, don’t think there is a relationship between the number of followers and time spent, you might write an if-then hypothesis that reads:

“If people have many followers on Instagram, they will not spend more time on the app than people who have less.”

In this case, the if-then suggests there isn’t a relationship between the variables. In that case, one of the null hypothesis examples might say:

“If people have many followers on Instagram, they will spend more time on the app than people who have less.”

You then test both the if-then and the null hypothesis to gauge if there is a relationship between the variables, and if so, how much of a relationship. 

4 Tips to Write the Best Hypothesis

If you’re going to take the time to hold an experiment, whether in school or by yourself, you’re also going to want to take the time to make sure your hypothesis is a good one. The best hypotheses have four major elements in common: plausibility, defined concepts, observability, and general explanation.

#1: Plausibility

At first glance, this quality of a hypothesis might seem obvious. When your hypothesis is plausible, that means it’s possible given what we know about science and general common sense. However, improbable hypotheses are more common than you might think. 

Imagine you’re studying weight gain and television watching habits. If you hypothesize that people who watch more than  twenty hours of television a week will gain two hundred pounds or more over the course of a year, this might be improbable (though it’s potentially possible). Consequently, c ommon sense can tell us the results of the study before the study even begins.

Improbable hypotheses generally go against  science, as well. Take this hypothesis example: 

“If a person smokes one cigarette a day, then they will have lungs just as healthy as the average person’s.” 

This hypothesis is obviously untrue, as studies have shown again and again that cigarettes negatively affect lung health. You must be careful that your hypotheses do not reflect your own personal opinion more than they do scientifically-supported findings. This plausibility points to the necessity of research before the hypothesis is written to make sure that your hypothesis has not already been disproven.

#2: Defined Concepts

The more advanced you are in your studies, the more likely that the terms you’re using in your hypothesis are specific to a limited set of knowledge. One of the hypothesis testing examples might include the readability of printed text in newspapers, where you might use words like “kerning” and “x-height.” Unless your readers have a background in graphic design, it’s likely that they won’t know what you mean by these terms. Thus, it’s important to either write what they mean in the hypothesis itself or in the report before the hypothesis.

Here’s what we mean. Which of the following sentences makes more sense to the common person?

If the kerning is greater than average, more words will be read per minute.

If the space between letters is greater than average, more words will be read per minute.

For people reading your report that are not experts in typography, simply adding a few more words will be helpful in clarifying exactly what the experiment is all about. It’s always a good idea to make your research and findings as accessible as possible. 

Good hypotheses ensure that you can observe the results. 

#3: Observability

In order to measure the truth or falsity of your hypothesis, you must be able to see your variables and the way they interact. For instance, if your hypothesis is that the flight patterns of satellites affect the strength of certain television signals, yet you don’t have a telescope to view the satellites or a television to monitor the signal strength, you cannot properly observe your hypothesis and thus cannot continue your study.

Some variables may seem easy to observe, but if you do not have a system of measurement in place, you cannot observe your hypothesis properly. Here’s an example: if you’re experimenting on the effect of healthy food on overall happiness, but you don’t have a way to monitor and measure what “overall happiness” means, your results will not reflect the truth. Monitoring how often someone smiles for a whole day is not reasonably observable, but having the participants state how happy they feel on a scale of one to ten is more observable. 

In writing your hypothesis, always keep in mind how you'll execute the experiment.

#4: Generalizability 

Perhaps you’d like to study what color your best friend wears the most often by observing and documenting the colors she wears each day of the week. This might be fun information for her and you to know, but beyond you two, there aren’t many people who could benefit from this experiment. When you start an experiment, you should note how generalizable your findings may be if they are confirmed. Generalizability is basically how common a particular phenomenon is to other people’s everyday life.

Let’s say you’re asking a question about the health benefits of eating an apple for one day only, you need to realize that the experiment may be too specific to be helpful. It does not help to explain a phenomenon that many people experience. If you find yourself with too specific of a hypothesis, go back to asking the big question: what is it that you want to know, and what do you think will happen between your two variables?

Hypothesis Testing Examples

We know it can be hard to write a good hypothesis unless you’ve seen some good hypothesis examples. We’ve included four hypothesis examples based on some made-up experiments. Use these as templates or launch pads for coming up with your own hypotheses.

Experiment #1: Students Studying Outside (Writing a Hypothesis)

You are a student at PrepScholar University. When you walk around campus, you notice that, when the temperature is above 60 degrees, more students study in the quad. You want to know when your fellow students are more likely to study outside. With this information, how do you make the best hypothesis possible?

You must remember to make additional observations and do secondary research before writing your hypothesis. In doing so, you notice that no one studies outside when it’s 75 degrees and raining, so this should be included in your experiment. Also, studies done on the topic beforehand suggested that students are more likely to study in temperatures less than 85 degrees. With this in mind, you feel confident that you can identify your variables and write your hypotheses:

If-then: “If the temperature in Fahrenheit is less than 60 degrees, significantly fewer students will study outside.”

Null: “If the temperature in Fahrenheit is less than 60 degrees, the same number of students will study outside as when it is more than 60 degrees.”

These hypotheses are plausible, as the temperatures are reasonably within the bounds of what is possible. The number of people in the quad is also easily observable. It is also not a phenomenon specific to only one person or at one time, but instead can explain a phenomenon for a broader group of people.

To complete this experiment, you pick the month of October to observe the quad. Every day (except on the days where it’s raining)from 3 to 4 PM, when most classes have released for the day, you observe how many people are on the quad. You measure how many people come  and how many leave. You also write down the temperature on the hour. 

After writing down all of your observations and putting them on a graph, you find that the most students study on the quad when it is 70 degrees outside, and that the number of students drops a lot once the temperature reaches 60 degrees or below. In this case, your research report would state that you accept or “failed to reject” your first hypothesis with your findings.

Experiment #2: The Cupcake Store (Forming a Simple Experiment)

Let’s say that you work at a bakery. You specialize in cupcakes, and you make only two colors of frosting: yellow and purple. You want to know what kind of customers are more likely to buy what kind of cupcake, so you set up an experiment. Your independent variable is the customer’s gender, and the dependent variable is the color of the frosting. What is an example of a hypothesis that might answer the question of this study?

Here’s what your hypotheses might look like: 

If-then: “If customers’ gender is female, then they will buy more yellow cupcakes than purple cupcakes.”

Null: “If customers’ gender is female, then they will be just as likely to buy purple cupcakes as yellow cupcakes.”

This is a pretty simple experiment! It passes the test of plausibility (there could easily be a difference), defined concepts (there’s nothing complicated about cupcakes!), observability (both color and gender can be easily observed), and general explanation ( this would potentially help you make better business decisions ).

Experiment #3: Backyard Bird Feeders (Integrating Multiple Variables and Rejecting the If-Then Hypothesis)

While watching your backyard bird feeder, you realized that different birds come on the days when you change the types of seeds. You decide that you want to see more cardinals in your backyard, so you decide to see what type of food they like the best and set up an experiment. 

However, one morning, you notice that, while some cardinals are present, blue jays are eating out of your backyard feeder filled with millet. You decide that, of all of the other birds, you would like to see the blue jays the least. This means you'll have more than one variable in your hypothesis. Your new hypotheses might look like this: 

If-then: “If sunflower seeds are placed in the bird feeders, then more cardinals will come than blue jays. If millet is placed in the bird feeders, then more blue jays will come than cardinals.”

Null: “If either sunflower seeds or millet are placed in the bird, equal numbers of cardinals and blue jays will come.”

Through simple observation, you actually find that cardinals come as often as blue jays when sunflower seeds or millet is in the bird feeder. In this case, you would reject your “if-then” hypothesis and “fail to reject” your null hypothesis . You cannot accept your first hypothesis, because it’s clearly not true. Instead you found that there was actually no relation between your different variables. Consequently, you would need to run more experiments with different variables to see if the new variables impact the results.

Experiment #4: In-Class Survey (Including an Alternative Hypothesis)

You’re about to give a speech in one of your classes about the importance of paying attention. You want to take this opportunity to test a hypothesis you’ve had for a while: 

If-then: If students sit in the first two rows of the classroom, then they will listen better than students who do not.

Null: If students sit in the first two rows of the classroom, then they will not listen better or worse than students who do not.

You give your speech and then ask your teacher if you can hand out a short survey to the class. On the survey, you’ve included questions about some of the topics you talked about. When you get back the results, you’re surprised to see that not only do the students in the first two rows not pay better attention, but they also scored worse than students in other parts of the classroom! Here, both your if-then and your null hypotheses are not representative of your findings. What do you do?

This is when you reject both your if-then and null hypotheses and instead create an alternative hypothesis . This type of hypothesis is used in the rare circumstance that neither of your hypotheses is able to capture your findings . Now you can use what you’ve learned to draft new hypotheses and test again! 

Key Takeaways: Hypothesis Writing

The more comfortable you become with writing hypotheses, the better they will become. The structure of hypotheses is flexible and may need to be changed depending on what topic you are studying. The most important thing to remember is the purpose of your hypothesis and the difference between the if-then and the null . From there, in forming your hypothesis, you should constantly be asking questions, making observations, doing secondary research, and considering your variables. After you have written your hypothesis, be sure to edit it so that it is plausible, clearly defined, observable, and helpful in explaining a general phenomenon.

Writing a hypothesis is something that everyone, from elementary school children competing in a science fair to professional scientists in a lab, needs to know how to do. Hypotheses are vital in experiments and in properly executing the scientific method . When done correctly, hypotheses will set up your studies for success and help you to understand the world a little better, one experiment at a time.

What’s Next?

If you’re studying for the science portion of the ACT, there’s definitely a lot you need to know. We’ve got the tools to help, though! Start by checking out our ultimate study guide for the ACT Science subject test. Once you read through that, be sure to download our recommended ACT Science practice tests , since they’re one of the most foolproof ways to improve your score. (And don’t forget to check out our expert guide book , too.)

If you love science and want to major in a scientific field, you should start preparing in high school . Here are the science classes you should take to set yourself up for success.

If you’re trying to think of science experiments you can do for class (or for a science fair!), here’s a list of 37 awesome science experiments you can do at home

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Ashley Sufflé Robinson has a Ph.D. in 19th Century English Literature. As a content writer for PrepScholar, Ashley is passionate about giving college-bound students the in-depth information they need to get into the school of their dreams.

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Why is proving the Riemann Hypothesis so hard?

The Riemann Hypothesis is considered by many to be the most important unsolved problem in pure mathematics.

Several attempts have been made in the last 150 years ( here some of them are reported ). RH is the only problem that has been listed in both Hilbert's 23 Problems and the Millennium Problems by the Clay Institute; and yet it remains unsolved, seemingly resisting all attacks, and quickly becoming a piece of mathematical folklore as an "impossible" problem. For instance, it is reported that Hilbert himself declared:

If I were to awaken after having slept for a thousand years, my first question would be: "Has the Riemann hypothesis been proven?"

Implications and applications of this possible result have already been addressed many times on this website (reporting here some of them): What does proving the Riemann Hypothesis accomplish? What is so interesting about the zeroes of the Riemann $\zeta$ function? Why do mathematicians care so much about zeta functions? What is the link between Primes and zeroes of Riemann zeta function?

What I'm asking is: why is a conjecture on the zeroes of a specific complex function so hard to prove (or disprove)? What are the main obstacles and obstructions to this problem's solution?

Edit: although the question has already been addressed before on Math.SE ( here and, in some way, here ), no answers have admittedly been given, and I personally find that the comments this question received in the last hours (for which I am grateful) addressed the problem much more clearly than the comments in the questions above. I believe there's room for improvement, but I apologize if the question is too broad or violates the guidelines in some other way.

• soft-question
• analytic-number-theory
• riemann-zeta
• riemann-hypothesis
• open-problem

• 6 $\begingroup$ The Riemann zeta function far from a simple object. On half of the complex plane, it's an infinite series. On the other half it is an analytic continuation of an infinite series, an even hairier object. We don't even half a general formula for the roots of degree 5 polynomials. Clearly the roots of an infinite series are a very elusive beast. $\endgroup$ –  K.defaoite Commented Dec 7, 2020 at 1:24
• 5 $\begingroup$ there are similar functions to RZ which have zeroes in the critical strip but not on the line showing that analytic properties like the functional equation are unlikely to suffice to prove RH if it's true; also RZ is a highly transcendental function (it is universal in a well-defined sense as one can approximate any non zero analytic function locally by values of RZ in the critical strip) and for example it sends any vertical line $1/2<\Re \sigma <1$ into a dense set in the plane etc $\endgroup$ –  Conrad Commented Dec 7, 2020 at 4:57
• 3 $\begingroup$ I'm voting to reopen since the linked question barely got any relevant answers. $\endgroup$ –  lisyarus Commented Dec 7, 2020 at 12:45
• 1 $\begingroup$ @lisyarus Isn't it likely the linked question did not receive relevant answers because the question is opinion based and possibly unanswerable? $\endgroup$ –  xxxxxxxxx Commented Mar 6, 2021 at 9:23
• 2 $\begingroup$ It's a survivor bias. We solve the easy problems pretty quickly. What's left? The hard ones. $\endgroup$ –  Asaf Karagila ♦ Commented Mar 6, 2021 at 11:06

This is a bit an opinion based question and answer.

The RH is about $\log\zeta(s),\frac1{\zeta(s)},\frac{\zeta'(s)}{\zeta(s)}$ , not $\zeta(s)$ .

On the $\zeta(s)$ side we can easily exploit that it is the Dirichlet series of the integers.

Surprisingly (or not?) on the $\log\zeta(s),\frac1{\zeta(s)},\frac{\zeta'(s)}{\zeta(s)}$ side we can't, and complicated structures appear, eg. the primes.

The same kind of structures appear for many other Dirichlet series (the Dirichlet L-functions, more generally the Selberg class) and the RH is (more or less) assumed to hold for all of them.

This set of Dirichlet series with a RH is discrete/isolated: you can't change slightly the coefficients without loosing one of the key properties (analytic continuation, functional equation, Euler product, growth of the coefficients).

So we need a setting where all those key properties are present: an arithmetical-analytical-algebraic setting . It is hard, by definition.

In practice most elementary approaches to the RH fail because they apply the same way to $(G(\chi_5)/5^{1/2})^{-1/2}L(s,\chi_5)+\overline{(G(\chi_5)/5^{1/2})}^{-1/2}L(s,\overline{\chi_5})$ lacking only the Euler product, and having a bunch of zeros off the critical line.

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Not the answer you're looking for browse other questions tagged soft-question analytic-number-theory riemann-zeta riemann-hypothesis open-problem ..

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No, autism is not caused by the gut microbiome.

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LONDON - JULY 28: London Zoo's "Dr Poo" gathers samples for London Zoo's Poo Exhibition at London ... [+] Zoo, Regent's Park on July 28, 2004 in London. Samples from Lamas to elephants are selected according to size, consistency, colour and smell in preparation for next week's 'Zoo Poo' exhibition, where visitors can discover how different animals poo and why. (Photo by Steve Finn/Getty Images)

The New York Times published a story last week claiming that we might be able to use the gut microbiome to diagnose autism. The Times story was based on a just-published scientific paper that claimed the same thing.

This report set off all my skeptical alarm bells. My initial reaction was “Oh no, more bad science around autism.” For one thing, as most scientists studying autism are aware, the modern anti-vaccine movement started with a scientific paper, back in 1998, that claimed, falsely, that childhood vaccines caused autism. That paper in The Lancet was later shown to be fraudulent and was eventually retracted, but not before a huge amount of damage was done. Its lead author, Andrew Wakefield, went on to become a hero to the anti-vaccine movement, and he continues to promote anti-vaccine misinformation to this day.

The new paper (from the journal Nature Microbiology ) is not making outrageous claims like that, nor was the New York Times . However, anyone claiming autism may be caused by microbes in the gut should know that the notorious Lancet study was based on a hypothesis about a “leaky gut,” a hypothesis that was discredited long ago. (I don’t want to give it any credibility, but that hypothesis held that virus particles in some vaccines somehow “leaked” from the gut and made their way to the brain. It was nonsense at the time and still is.) That’s one reason why the suggestion that microbes in the gut might be used to diagnose autism raises so many alarm bells.

I’ve now looked at the study, and frankly I am deeply skeptical. Let me be clear, though: I’m not trying to prove scientifically that the study is wrong, which would require many months of effort and a much more detail than I can put into a column anyway. Fortunately, though, there’s an earlier study that did that job for me, which I’ll get to below.

However, the science behind this study is closely related to my own work, so I feel pretty comfortable offering my expert perspective. So what did the authors do?

Well, as the new study explains, they collected poop (“faecal samples”) from 1,627 children, some of whom had been diagnosed with autism and some who hadn’t, and they sequenced DNA from the poop. Then they looked for bacteria, viruses, and other microbes in the DNA sequence data.

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That’s right: the “gut microbiome,” is really just a polite term for bacteria that live in the intestines and the colon, some of which come out in poop. Of course, some bacteria in poop might come from the food that a person ate, but mostly these are so-called gut bacteria.

I’ve been involved in many studies like this myself, so I’ve seen that these experiments yield hundreds of different species from every sample. The data sets are very complex, and a widespread problem in the field is that these data are often misinterpreted. In the Nature Microbiology paper, the authors took these very complex data sets and fed them to a machine learning program, and voila! The AI program was able to do a pretty good job (far from perfect, I should note) identifying the autistic children, based on the melange of microbes in their poop.

So what’s the problem? Well, first of all, machine learning programs are really good at telling apart two sets of subjects (such as children with and without autism) if you give them enough data. It sometimes turns out that the learning programs are keying in on irrelevant features that the scientists didn’t intend.

For example, this 2021 paper looked at over 400 studies that used machine learning to predict Covid-19, all of which had claimed some success, and found that all of the studies were essentially useless “due to methodological flaws and/or underlying biases.” Of course, the gut microbiome study wasn’t one of those, and some machine learning experiments do work, but we should be very skeptical.

Another reason for skepticism is that the new paper doesn’t even try to tell us what the machine learning models actually learned–it just treats the programs as a “black box” that we should trust.

Perhaps the biggest flaw in the study of autism and children’s gut microbiota is this: children with autism tend to be finicky eaters, and their parents try all sorts of diets in the hope that they can at least alleviate the symptoms of autism with food. There are countless websites–many of them scams, unfortunately–claiming that special diets can help these children. Why is this important? Because a special diet will alter your gut microbiome, sometimes quite significantly.

Thus even if the machine learning models in the new study are correct, the causality almost certainly goes the other way: children with autism might have a different microbiome because they’re eating different foods. Thus it’s autism that indirectly affects the microbiome. Unfortunately, both the New York Times and the scientific paper suggested the opposite: for example, the Times paraphrases the scientists stating that “gut bacteria, fungi, viruses and more could one day be the basis of a diagnostic tool” for autism.

Now on to that earlier scientific paper I mentioned above. It turns out that three years ago, a group of researchers in Australia published a major study in the journal Cell that addressed precisely the problem I just pointed out. In that study, the scientists collected and sequenced poop from 247 children both with and without autism. They found “negligible direct associations between ASD [autism spectrum disorder] and the gut microbiome.”

On the contrary, the authors warned: “microbiome differences in ASD may reflect dietary preferences ... and we caution against claims that the microbiome has a driving role in ASD.”

In other words, three years ago a study in a major scientific journal found that there was no connection between autism and the contents of the gut microbiome. They went on to warn that if you see differences in the gut microbiome in autistic kids, those are caused by their diet, so don’t go claiming that the microbiome causes autism. The authors of the newer study, and the reporters at the New York Times , apparently decided otherwise.

So no, the evidence doesn’t support the claim that the gut microbiome can be used to diagnose autism.

Update: I’ve reached out to the authors of the study and to the New York Times for comment. This article will be updated if they respond.

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What Is Project 2025, and Why Is Trump Disavowing It?

The Biden campaign has attacked Donald J. Trump’s ties to the conservative policy plan that would amass power in the executive branch, though it is not his official platform.

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Donald J. Trump has gone to great lengths to distance himself from Project 2025, a set of conservative policy proposals for a future Republican administration that has outraged Democrats. He has claimed he knows nothing about it or the people involved in creating it.

Mr. Trump himself was not behind the project. But some of his allies were.

The document, its origins and the interplay between it and the Trump campaign have made for one of the most hotly debated questions of the 2024 race.

Here is what to know about Project 2025, and who is behind it.

What is Project 2025?

Project 2025 was spearheaded by the Heritage Foundation and like-minded conservative groups before Mr. Trump officially entered the 2024 race. The Heritage Foundation is a think tank that has shaped the personnel and policies of Republican administrations since the Reagan presidency.

The project was intended as a buffet of options for the Trump administration or any other Republican presidency. It’s the latest installment in the Heritage Foundation’s Mandate for Leadership series, which has compiled conservative policy proposals every few years since 1981. But no previous study has been as sweeping in its recommendations — or as widely discussed.

Kevin Roberts, the head of the Heritage Foundation, which began putting together the latest document in 2022, said he thought the American government would embrace a more conservative era, one that he hoped Republicans would usher in.

“We are in the process of the second American Revolution,” Mr. Roberts said on Real America’s Voice, a right-wing cable channel, in early July, adding pointedly that the revolt “will remain bloodless if the left allows it to be.”

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What we know about the Trump assassination attempt and the shooter

Former President Donald Trump was shot during a rally in Pennsylvania on Saturday. He was immediately shielded and taken to safety after his ear was injured. Details are emerging about how it happened and the identity of the shooter, Thomas Matthew Crooks , 20.

President Joe Biden delivered an address from the Oval Office on Sunday night and said he was grateful Trump was not seriously hurt.

Here’s what we know.

Where did it happen?

The shooting took place at a presidential campaign rally in Butler, Pennsylvania , a city in the western part of the state about an hour's drive north of Pittsburgh.

What happened?

About six minutes into Trump’s speech, the former president could be seen clutching his ear after popping noises rang out over the rally. Trump ducked to the ground as several Secret Service agents rushed to the stage and surrounded him on all sides. There were screams from onlookers as the scene unfolded.

Roughly a minute later, agents helped Trump get up from the ground and stand. He held up his fist to the crowd, prompting cheers from supporters. Several agents then rushed him off the stage and escorted him into a vehicle.

Blood could be seen on Trump’s ear and on the side of his face. He later said on his social media site, Truth Social, that a bullet “pierced the upper part of my right ear.”

According to preliminary reports, which could change as the crime scene is processed, eight shots were fired by the shooter, an official said.

Trump was talking about Biden’s policies on immigration when shots were fired.

Was anyone else hurt? 

One spectator died and two others were critically wounded, Pennsylvania State Police Lt. Col. George Bivens said.

The man who died was 50-year-old former firefighter Corey Comperatore, Pennsylvania Gov. Josh Shapiro said Sunday. He said that Comperatore, whom he called a hero, moved to protect his wife and two daughters when gunshots were heard at the rally.

The two people who were wounded were David Dutch, 57, of New Kensington, Pennsylvania, and James Copenhaver, 74, of Moon Township, Pennsylvania. Both are being treated at Allegheny General Hospital in Pittsburgh and are listed as being in stable condition.

Witnesses described hearing loud popping noises, with one person saying they saw someone who was shot in the back of the head and another person who said she saw someone “bleeding profusely.”

Rep. Ronny Jackson, R-Texas, said in a statement that his nephew was injured in the shooting. Jackson said that his family was seated near Trump and that after they heard shots, his nephew realized something had grazed and cut his neck.

He said his nephew’s injury was not serious and “he is doing well.”

Was it an assassination attempt? Which agencies are investigating?

Law enforcement officials are investigating the incident as an assassination attempt.

“This evening, we had what we’re calling an assassination attempt against our former president, Donald Trump,” said Kevin Rojek, special agent in charge of the FBI’s Pittsburgh field office.

The FBI is investigating it as an act of domestic terrorism. The agency is leading efforts and working alongside the Secret Service and state and local law enforcement. In the aftermath of the shooting, the FBI deployed investigative agents, bomb technicians and evidence response personnel.

Rojek asked that witnesses to the shooting contact the FBI.

Who is the shooter? What was the motive?

The FBI identified the shooter as 20-year-old Thomas Matthew Crooks of Bethel Park, Pennsylvania. He was killed at the scene.

“We do not currently have an identified motive, although our investigators are working tirelessly to attempt to identify what that motive was,” Rojek said.

A senior law enforcement official directly briefed on the matter said the nature of the shooting suggests political ideology as motive but that there was nothing definitive known at the time. The FBI conducted a preliminary analysis of Crooks’ phone but has not found anything to indicate motive, according to a senior U.S. law enforcement official.

Crooks used a semiautomatic rifle, based on what was found at the scene, three senior U.S. law enforcement officials said. FBI officials said that the weapon he used is believed to have been bought by his father, though they don’t have any additional information right now on how he got it. More than a dozen guns were also found in a search of the Crooks family home, according to four senior officials. The family is cooperating with investigators, an official said.

Law enforcement officials found a number of suspicious canisters or containers in Crooks’ vehicle, which was left near the rally, but it was unclear if they were functional as incendiary or explosive devices, two officials said.

The shooter was part of an area gun club, the Clairton Sportsmen’s Club. At its range in Clairton, the club has facilities for skeet shooting, high-power rifle exercises and archery practice.

Voter records from Pennsylvania listed a person with the same name, address and birth date as a registered Republican.

He did not have any affiliation with the U.S. military, according to the Defense Department.

It was not known whether the shooter was acting alone or in coordination with others. There does not appear to be any evidence currently that the shooting had any link to a foreign actor, according to a U.S. official.

Where was the shooter? Did the shooter get past security?

Crooks fired several shots from a nearby rooftop during the rally. The rooftop was outside the security perimeter established by the Secret Service, three senior law enforcement officials told NBC News.

Rallygoers alerted local police of a suspicious person near the magnetometer area, two senior officials said. They tried to search for the suspicious person, believed to be Crooks, but could not find him in the crowd, the officials said.

Two municipal officers also tried to approach Crooks shortly before he opened fire, according to two senior officials. It is unclear where that took place.

Crooks’ father called the police after the shooting to say he was worried that his son and his AR rifle were missing, according to three senior officials.

What have we heard from Trump?

Trump first posted about the incident on Truth Social.

“I was shot with a bullet that pierced the upper part of my right ear. I knew immediately that something was wrong in that I heard a whizzing sound, shots, and immediately felt the bullet ripping through the skin. Much bleeding took place, so I realized then what was happening. GOD BLESS AMERICA!” he wrote.

He expressed gratitude to the Secret Service and law enforcement for their quick response. He added, “Most importantly, I want to extend my condolences to the family of the person at the Rally who was killed, and also to the family of another person that was badly injured.”

“It is incredible that such an act can take place in our Country. Nothing is known at this time about the shooter, who is now dead,” he wrote.

What have we heard from Trump’s family?

Melania Trump released a statement Sunday describing the fear she felt seeing the incident unfold and thanking the Secret Service agents and law enforcement who responded. She also offered sympathy for the victims.

“A monster who recognized my husband as an inhuman political machine attempted to ring out Donald’s passion — his laughter, ingenuity, love of music, and inspiration,” she wrote in the statement. “The core facets of my husband’s life — his human side — were buried below the political machine.”

She encouraged people to look beyond partisan politics: “Let us not forget that differing opinions, policy, and political games are inferior to love.”

Trump’s eldest daughter, Ivanka Trump, condemned the attack in a statement on X , thanking people for their prayers. “I am grateful to the Secret Service and all the other law enforcement officers for their quick and decisive actions today,” she wrote. “I continue to pray for our country.”

“I love you Dad, today and always,” she added.

Trump’s eldest son, Donald Trump Jr., said in a statement through his spokesperson that he spoke to his father over the phone Saturday and that he is in “great spirits.” “He will never stop fighting to save America, no matter what the radical left throws at him,” Trump Jr. said.

Their brother Eric Trump posted on X a picture of his father pumping his fist in the air after the shooting. “This is the fighter America needs!” the post read.

Has Biden responded?

In his address from the Oval Office , Biden shared his condolences to those who were hurt and championed the values of democracy. "The higher the stakes, the more fervent the passions become," he said. "This places an added burden on each of us to ensure that no matter how strong, our convictions must never descend into violence."

The president also held a news conference earlier in the day, in which he said he'd spoken with Trump, who was "doing well and recovering." Biden added that he and Vice President Kamala Harris have been briefed by law enforcement, the national security adviser and Homeland Security.

Biden said he has directed the Secret Service to provide Trump with "every resource capability and protective measure necessary to ensure his continued safety." He has also asked the Secret Service to review security for the Republican National Convention, which begins Monday, and ordered an independent review of national security at the rally.

The president urged unity and cautioned against assumptions about the suspect's motives.

Biden first addressed reporters about the shooting from Rehoboth Beach, Delaware, about 90 minutes after it occurred.

“It’s sick. It’s sick. It’s one of the reasons we have to unite this country,” he said. “We cannot condone this.”

The president's campaigning was put on hold for roughly 36 hours and began again Monday afternoon.

Matt Lavietes is a reporter for NBC Out.