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Published on January 28, 2020 by Rebecca Bevans . Revised on June 22, 2023.
Statistical tests are used in hypothesis testing . They can be used to:
Statistical tests assume a null hypothesis of no relationship or no difference between groups. Then they determine whether the observed data fall outside of the range of values predicted by the null hypothesis.
If you already know what types of variables you’re dealing with, you can use the flowchart to choose the right statistical test for your data.
Statistical tests flowchart
What does a statistical test do, when to perform a statistical test, choosing a parametric test: regression, comparison, or correlation, choosing a nonparametric test, flowchart: choosing a statistical test, other interesting articles, frequently asked questions about statistical tests.
Statistical tests work by calculating a test statistic – a number that describes how much the relationship between variables in your test differs from the null hypothesis of no relationship.
It then calculates a p value (probability value). The p -value estimates how likely it is that you would see the difference described by the test statistic if the null hypothesis of no relationship were true.
If the value of the test statistic is more extreme than the statistic calculated from the null hypothesis, then you can infer a statistically significant relationship between the predictor and outcome variables.
If the value of the test statistic is less extreme than the one calculated from the null hypothesis, then you can infer no statistically significant relationship between the predictor and outcome variables.
You can perform statistical tests on data that have been collected in a statistically valid manner – either through an experiment , or through observations made using probability sampling methods .
For a statistical test to be valid , your sample size needs to be large enough to approximate the true distribution of the population being studied.
To determine which statistical test to use, you need to know:
Statistical tests make some common assumptions about the data they are testing:
If your data do not meet the assumptions of normality or homogeneity of variance, you may be able to perform a nonparametric statistical test , which allows you to make comparisons without any assumptions about the data distribution.
If your data do not meet the assumption of independence of observations, you may be able to use a test that accounts for structure in your data (repeated-measures tests or tests that include blocking variables).
The types of variables you have usually determine what type of statistical test you can use.
Quantitative variables represent amounts of things (e.g. the number of trees in a forest). Types of quantitative variables include:
Categorical variables represent groupings of things (e.g. the different tree species in a forest). Types of categorical variables include:
Choose the test that fits the types of predictor and outcome variables you have collected (if you are doing an experiment , these are the independent and dependent variables ). Consult the tables below to see which test best matches your variables.
Parametric tests usually have stricter requirements than nonparametric tests, and are able to make stronger inferences from the data. They can only be conducted with data that adheres to the common assumptions of statistical tests.
The most common types of parametric test include regression tests, comparison tests, and correlation tests.
Regression tests look for cause-and-effect relationships . They can be used to estimate the effect of one or more continuous variables on another variable.
Predictor variable | Outcome variable | Research question example | |
---|---|---|---|
What is the effect of income on longevity? | |||
What is the effect of income and minutes of exercise per day on longevity? | |||
Logistic regression | What is the effect of drug dosage on the survival of a test subject? |
Comparison tests look for differences among group means . They can be used to test the effect of a categorical variable on the mean value of some other characteristic.
T-tests are used when comparing the means of precisely two groups (e.g., the average heights of men and women). ANOVA and MANOVA tests are used when comparing the means of more than two groups (e.g., the average heights of children, teenagers, and adults).
Predictor variable | Outcome variable | Research question example | |
---|---|---|---|
Paired t-test | What is the effect of two different test prep programs on the average exam scores for students from the same class? | ||
Independent t-test | What is the difference in average exam scores for students from two different schools? | ||
ANOVA | What is the difference in average pain levels among post-surgical patients given three different painkillers? | ||
MANOVA | What is the effect of flower species on petal length, petal width, and stem length? |
Correlation tests check whether variables are related without hypothesizing a cause-and-effect relationship.
These can be used to test whether two variables you want to use in (for example) a multiple regression test are autocorrelated.
Variables | Research question example | |
---|---|---|
Pearson’s | How are latitude and temperature related? |
Non-parametric tests don’t make as many assumptions about the data, and are useful when one or more of the common statistical assumptions are violated. However, the inferences they make aren’t as strong as with parametric tests.
Predictor variable | Outcome variable | Use in place of… | |
---|---|---|---|
Spearman’s | |||
Pearson’s | |||
Sign test | One-sample -test | ||
Kruskal–Wallis | ANOVA | ||
ANOSIM | MANOVA | ||
Wilcoxon Rank-Sum test | Independent t-test | ||
Wilcoxon Signed-rank test | Paired t-test | ||
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This flowchart helps you choose among parametric tests. For nonparametric alternatives, check the table above.
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
Methodology
Research bias
Statistical tests commonly assume that:
If your data does not meet these assumptions you might still be able to use a nonparametric statistical test , which have fewer requirements but also make weaker inferences.
A test statistic is a number calculated by a statistical test . It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.
The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.
Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.
Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .
When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.
Quantitative variables are any variables where the data represent amounts (e.g. height, weight, or age).
Categorical variables are any variables where the data represent groups. This includes rankings (e.g. finishing places in a race), classifications (e.g. brands of cereal), and binary outcomes (e.g. coin flips).
You need to know what type of variables you are working with to choose the right statistical test for your data and interpret your results .
Discrete and continuous variables are two types of quantitative variables :
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The bottom line.
Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.
Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population or a data-generating process. The word "population" will be used for both of these cases in the following descriptions.
In hypothesis testing, an analyst tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.
The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis. Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.
The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.
If an individual wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct. Mathematically, the null hypothesis is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.
A random sample of 100 coin flips is taken, and the null hypothesis is tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.
If there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."
Some statisticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”
Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.
Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.
Hypothesis testing refers to a statistical process that helps researchers determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. All hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.
Sage. " Introduction to Hypothesis Testing ," Page 4.
Elder Research. " Who Invented the Null Hypothesis? "
Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ."
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Hypothesis testing in statistics involves testing an assumption about a population parameter using sample data. Learners can download Hypothesis Testing PDF to get instant access to all information!
What exactly is hypothesis testing, and how does it work in statistics? Can I find practical examples and understand the different types from this blog?
Hypothesis Testing : Ever wonder how researchers determine if a new medicine actually works or if a new marketing campaign effectively drives sales? They use hypothesis testing! It is at the core of how scientific studies, business experiments and surveys determine if their results are statistically significant or just due to chance.
Hypothesis testing allows us to make evidence-based decisions by quantifying uncertainty and providing a structured process to make data-driven conclusions rather than guessing. In this post, we will discuss hypothesis testing types, examples, and processes!
Table of Contents
Hypothesis testing is a statistical method used to evaluate the validity of a hypothesis using sample data. It involves assessing whether observed data provide enough evidence to reject a specific hypothesis about a population parameter.
Hypothesis testing in data science is a statistical method used to evaluate two mutually exclusive population statements based on sample data. The primary goal is to determine which statement is more supported by the observed data.
Hypothesis testing assists in supporting the certainty of findings in research and data science projects. This statistical inference aids in making decisions about population parameters using sample data. For those who are looking to deepen their knowledge in data science and expand their skillset, we highly recommend checking out Master Generative AI: Data Science Course by Physics Wallah .
Also Read: What is Encapsulation Explain in Details
The hypothesis testing procedure in data science involves a structured approach to evaluating hypotheses using statistical methods. Here’s a step-by-step breakdown of the typical procedure:
Also Read: Binary Search Algorithm
Hypothesis testing is a fundamental concept in statistics that aids analysts in making informed decisions based on sample data about a larger population. The process involves setting up two contrasting hypotheses, the null hypothesis and the alternative hypothesis, and then using statistical methods to determine which hypothesis provides a more plausible explanation for the observed data.
Once these hypotheses are established, analysts gather data from a sample and conduct statistical tests. The objective is to determine whether the observed results are statistically significant enough to reject the null hypothesis in favor of the alternative.
Hypothesis testing is a cornerstone in statistical analysis, providing a framework to evaluate the validity of assumptions or claims made about a population based on sample data. Within this framework, several specific tests are utilized based on the nature of the data and the question at hand. Here’s a closer look at the three fundamental types of hypothesis tests:
The z-test is a statistical method primarily employed when comparing means from two datasets, particularly when the population standard deviation is known. Its main objective is to ascertain if the means are statistically equivalent.
A crucial prerequisite for the z-test is that the sample size should be relatively large, typically 30 data points or more. This test aids researchers and analysts in determining the significance of a relationship or discovery, especially in scenarios where the data’s characteristics align with the assumptions of the z-test.
The t-test is a versatile statistical tool used extensively in research and various fields to compare means between two groups. It’s particularly valuable when the population standard deviation is unknown or when dealing with smaller sample sizes.
By evaluating the means of two groups, the t-test helps ascertain if a particular treatment, intervention, or variable significantly impacts the population under study. Its flexibility and robustness make it a go-to method in scenarios ranging from medical research to business analytics.
The Chi-Square test stands distinct from the previous tests, primarily focusing on categorical data rather than means. This statistical test is instrumental when analyzing categorical variables to determine if observed data aligns with expected outcomes as posited by the null hypothesis.
By assessing the differences between observed and expected frequencies within categorical data, the Chi-Square test offers insights into whether discrepancies are statistically significant. Whether used in social sciences to evaluate survey responses or in quality control to assess product defects, the Chi-Square test remains pivotal for hypothesis testing in diverse scenarios.
Also Read: Python vs Java: Which is Best for Machine learning algorithm
Hypothesis testing is a fundamental concept in statistics used to make decisions or inferences about a population based on a sample of data. The process involves setting up two competing hypotheses, the null hypothesis H 0 and the alternative hypothesis H 1.
Through various statistical tests, such as the t-test, z-test, or Chi-square test, analysts evaluate sample data to determine whether there’s enough evidence to reject the null hypothesis in favor of the alternative. The aim is to draw conclusions about population parameters or to test theories, claims, or hypotheses.
In research, hypothesis testing serves as a structured approach to validate or refute theories or claims. Researchers formulate a clear hypothesis based on existing literature or preliminary observations. They then collect data through experiments, surveys, or observational studies.
Using statistical methods, researchers analyze this data to determine if there’s sufficient evidence to reject the null hypothesis. By doing so, they can draw meaningful conclusions, make predictions, or recommend actions based on empirical evidence rather than mere speculation.
R, a powerful programming language and environment for statistical computing and graphics, offers a wide array of functions and packages specifically designed for hypothesis testing. Here’s how hypothesis testing is conducted in R:
Hypothesis testing is an integral part of statistics and research, offering a systematic approach to validate hypotheses. Leveraging R’s capabilities, researchers and analysts can efficiently conduct and interpret various hypothesis tests, ensuring robust and reliable conclusions from their data.
Yes, data scientists frequently engage in hypothesis testing as part of their analytical toolkit. Hypothesis testing is a foundational statistical technique used to make data-driven decisions, validate assumptions, and draw conclusions from data. Here’s how data scientists utilize hypothesis testing:
Let’s delve into some common examples of hypothesis testing and provide solutions or interpretations for each scenario.
Scenario : A coffee shop owner believes that the average waiting time for customers during peak hours is 5 minutes. To test this, the owner takes a random sample of 30 customer waiting times and wants to determine if the average waiting time is indeed 5 minutes.
Hypotheses :
Solution : Using a t-test (assuming population variance is unknown), calculate the t-statistic based on the sample mean, sample standard deviation, and sample size. Then, determine the p-value and compare it with a significance level (e.g., 0.05) to decide whether to reject the null hypothesis.
Scenario : An e-commerce company wants to determine if changing the color of a “Buy Now” button from blue to green increases the conversion rate.
Solution : Split website visitors into two groups: one sees the blue button (control group), and the other sees the green button (test group). Track the conversion rates for both groups over a specified period. Then, use a chi-square test or z-test (for large sample sizes) to determine if there’s a statistically significant difference in conversion rates between the two groups.
The formula for hypothesis testing typically depends on the type of test (e.g., z-test, t-test, chi-square test) and the nature of the data (e.g., mean, proportion, variance). Below are the basic formulas for some common hypothesis tests:
Z-Test for Population Mean :
Z=(σ/n)(xˉ−μ0)
T-Test for Population Mean :
t= (s/ n ) ( x ˉ −μ 0 )
s = Sample standard deviation
Chi-Square Test for Goodness of Fit :
χ2=∑Ei(Oi−Ei)2
Also Read: Full Form of OOPS
While you can perform hypothesis testing manually using the above formulas and statistical tables, many online tools and software packages simplify this process. Here’s how you might use a calculator or software:
When using any calculator or software, always ensure you understand the underlying assumptions of the test, interpret the results correctly, and consider the broader context of your research or analysis.
What are the key components of a hypothesis test.
The key components include: Null Hypothesis (H0): A statement of no effect or no difference. Alternative Hypothesis (H1 or Ha): A statement that contradicts the null hypothesis. Test Statistic: A value computed from the sample data to test the null hypothesis. Significance Level (α): The threshold for rejecting the null hypothesis. P-value: The probability of observing the given data, assuming the null hypothesis is true.
The significance level (often denoted as α) is the probability threshold used to determine whether to reject the null hypothesis. Commonly used values for α include 0.05, 0.01, and 0.10, representing a 5%, 1%, or 10% chance of rejecting the null hypothesis when it's actually true.
The choice between one-tailed and two-tailed tests depends on your research question and hypothesis. Use a one-tailed test when you're specifically interested in one direction of an effect (e.g., greater than or less than). Use a two-tailed test when you want to determine if there's a significant difference in either direction.
The p-value is a probability value that helps determine the strength of evidence against the null hypothesis. A low p-value (typically ≤ 0.05) suggests that the observed data is inconsistent with the null hypothesis, leading to its rejection. Conversely, a high p-value suggests that the data is consistent with the null hypothesis, leading to no rejection.
No, hypothesis testing cannot prove a hypothesis true. Instead, it helps assess the likelihood of observing a given set of data under the assumption that the null hypothesis is true. Based on this assessment, you either reject or fail to reject the null hypothesis.
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Here is a list hypothesis testing exercises and solutions. Try to solve a question by yourself first before you look at the solution.
Question 1 In the population, the average IQ is 100 with a standard deviation of 15. A team of scientists want to test a new medication to see if it has either a positive or negative effect on intelligence, or not effect at all. A sample of 30 participants who have taken the medication has a mean of 140. Did the medication affect intelligence? View Solution to Question 1
A professor wants to know if her introductory statistics class has a good grasp of basic math. Six students are chosen at random from the class and given a math proficiency test. The professor wants the class to be able to score above 70 on the test. The six students get the following scores:62, 92, 75, 68, 83, 95. Can the professor have 90% confidence that the mean score for the class on the test would be above 70. Solution to Question 2
Question 3 In a packaging plant, a machine packs cartons with jars. It is supposed that a new machine would pack faster on the average than the machine currently used. To test the hypothesis, the time it takes each machine to pack ten cartons are recorded. The result in seconds is as follows.
42.1 | 42.7 |
41 | 43.6 |
41.3 | 43.8 |
41.8 | 43.3 |
42.4 | 42.5 |
42.8 | 43.5 |
43.2 | 43.1 |
42.3 | 41.7 |
41.8 | 44 |
42.7 | 44.1 |
Do the data provide sufficient evidence to conclude that, on the average, the new machine packs faster? Perform the required hypothesis test at the 5% level of significance. Solution to Question 3
Question 4 We want to compare the heights in inches of two groups of individuals. Here are the measurements: X: 175, 168, 168, 190, 156, 181, 182, 175, 174, 179 Y: 120, 180, 125, 188, 130, 190, 110, 185, 112, 188 Solution to Question 4
Question 5 A clinic provides a program to help their clients lose weight and asks a consumer agency to investigate the effectiveness of the program. The agency takes a sample of 15 people, weighing each person in the sample before the program begins and 3 months later. The results a tabulated below
Determine is the program is effective. Solution to Question 5
Question 6 A sample of 20 students were selected and given a diagnostic module prior to studying for a test. And then they were given the test again after completing the module. . The result of the students scores in the test before and after the test is tabulated below.
We want to see if there is significant improvement in the student’s performance due to this teaching method Solution to Question 6
Question 7 A study was performed to test wether cars get better mileage on premium gas than on regular gas. Each of 10 cars was first filled with regular or premium gas, decided by a coin toss, and the mileage for the tank was recorded. The mileage was recorded again for the same cars using other kind of gasoline. Determine wether cars get significantly better mileage with premium gas.
Mileage with regular gas: 16,20,21,22,23,22,27,25,27,28 Mileage with premium gas: 19, 22,24,24,25,25,26,26,28,32 Solution to Question 7
Question 8 An automatic cutter machine must cut steel strips of 1200 mm length. From a preliminary data, we checked that the lengths of the pieces produced by the machine can be considered as normal random variables with a 3mm standard deviation. We want to make sure that the machine is set correctly. Therefore 16 pieces of the products are randomly selected and weight. The figures were in mm: 1193,1196,1198,1195,1198,1199,1204,1193,1203,1201,1196,1200,1191,1196,1198,1191 Examine wether there is any significant deviation from the required size Solution to Question 8
Question 9 Blood pressure reading of ten patients before and after medication for reducing the blood pressure are as follows
Patient: 1,2,3,4,5,6,7,8,9,10 Before treatment: 86,84,78,90,92,77,89,90,90,86 After treatment: 80,80,92,79,92,82,88,89,92,83
Test the null hypothesis of no effect agains the alternate hypothesis that medication is effective. Execute it with Wilcoxon test Solution to Question 9
Question on ANOVA Sussan Sound predicts that students will learn most effectively with a constant background sound, as opposed to an unpredictable sound or no sound at all. She randomly divides 24 students into three groups of 8 each. All students study a passage of text for 30 minutes. Those in group 1 study with background sound at a constant volume in the background. Those in group 2 study with nose that changes volume periodically. Those in group 3 study with no sound at all. After studying, all students take a 10 point multiple choice test over the material. Their scores are tabulated below.
Group1: Constant sound: 7,4,6,8,6,6,2,9 Group 2: Random sound: 5,5,3,4,4,7,2,2 Group 3: No sound at all: 2,4,7,1,2,1,5,5 Solution to Question 10
Question 11 Using the following three groups of data, perform a one-way analysis of variance using α = 0.05.
51 | 23 | 56 |
45 | 43 | 76 |
33 | 23 | 74 |
45 | 43 | 87 |
67 | 45 | 56 |
Solution to Question 11
Question 12 In a packaging plant, a machine packs cartons with jars. It is supposed that a new machine would pack faster on the average than the machine currently used. To test the hypothesis, the time it takes each machine to pack ten cartons are recorded. The result in seconds is as follows.
New Machine: 42,41,41.3,41.8,42.4,42.8,43.2,42.3,41.8,42.7 Old Machine: 42.7,43.6,43.8,43.3,42.5,43.5,43.1,41.7,44,44.1
Perform an F-test to determine if the null hypothesis should be accepted. Solution to Question 12
Question 13 A random sample 500 U.S adults are questioned about their political affiliation and opinion on a tax reform bill. We need to test if the political affiliation and their opinon on a tax reform bill are dependent, at 5% level of significance. The observed contingency table is given below.
total | ||||
138 | 83 | 64 | 285 | |
64 | 67 | 84 | 215 | |
total | 202 | 150 | 148 | 500 |
Solution to Question 13
Question 14 Can a dice be considered regular which is showing the following frequency distribution during 1000 throws?
1 | 2 | 3 | 4 | 5 | 6 | |
182 | 154 | 162 | 175 | 151 | 176 |
Solution to Question 14
Solution to Question 15
Question 16 A newly developed muesli contains five types of seeds (A, B, C, D and E). The percentage of which is 35%, 25%, 20%, 10% and 10% according to the product information. In a randomly selected muesli, the following volume distribution was found.
Component | A | B | C | D | E |
Number of Pieces | 184 | 145 | 100 | 63 | 63 |
Lets us decide about the null hypothesis whether the composition of the sample corresponds to the distribution indicated on the packaging at alpha = 0.1 significance level. Solution to Question 16
Question 17 A research team investigated whether there was any significant correlation between the severity of a certain disease runoff and the age of the patients. During the study, data for n = 200 patients were collected and grouped according to the severity of the disease and the age of the patient. The table below shows the result
41 | 34 | 9 | ||
25 | 25 | 12 | ||
6 | 33 | 15 |
Let us decided about the correlation between the age of the patients and the severity of disease progression. Solution to Question 17
Question 18 A publisher is interested in determine which of three book cover is most attractive. He interviews 400 people in each of the three states (California, Illinois and New York), and asks each person which of the cover he or she prefers. The number of preference for each cover is as follows:
81 | 60 | 182 | 323 | |
78 | 93 | 95 | 266 | |
241 | 247 | 123 | 611 | |
400 | 400 | 400 | 1200 |
Do these data indicate that there are regional differences in people’s preferences concerning these covers? Use the 0.05 level of significance. Solution to Question 18
Question 19 Trees planted along the road were checked for which ones are healthy(H) or diseased (D) and the following arrangement of the trees were obtained:
H H H H D D D H H H H H H H D D H H D D D
Test at the = 0.05 significance wether this arrangement may be regarded as random
Solution to Question 19
Question 20 Suppose we flip a coin n = 15 times and come up with the following arrangements
H T T T H H T T T T H H T H H
(H = head, T = tail)
Test at the alpha = 0.05 significance level whether this arrangement may be regarded as random.
Solution to Question 20
You might also like, chi-square test for independence – question 18 (a publisher is interested…), hypothesis testing question 19 – run test ( trees were planted…), hypothesis testing question 21 – wald-wolfowitz run test for large sample (step by step procedure).
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Below are given the gain in weights (in lbs.) of pigs fed on two diet A and B Dieta 25 32 30 34 24 14 32 24 30 31 35 25 – – DietB 44 34 22 10 47 31 40 30 32 35 18 21 35 29
Statistics By Jim
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The Kruskal Wallis test is a nonparametric hypothesis test that compares three or more independent groups. Statisticians also refer to it as one-way ANOVA on ranks. This analysis extends the Mann Whitney U nonparametric test that can compare only two groups.
If you analyze data, chances are you’re familiar with one-way ANOVA that compares the means of at least three groups. The Kruskal Wallis test is the nonparametric version of it. Because it is nonparametric, the analysis makes fewer assumptions about your data than its parametric equivalent.
Many analysts use the Kruskal Wallis test to determine whether the medians of at least three groups are unequal. However, it’s important to note that it only assesses the medians in particular circumstances. Interpreting the analysis results can be thorny. More on this later!
If you need a nonparametric test for paired groups or a single sample , consider the Wilcoxon signed rank test .
Learn more about Parametric vs. Nonparametric Tests and Hypothesis Testing Overview .
At its core, the Kruskal Wallis test evaluates data ranks. The procedure ranks all the sample data from low to high. Then it averages the ranks for all groups. If the results are statistically significant, the average group ranks are not all equal. Consequently, the analysis indicates whether any groups have values that rank differently. For instance, one group might have values that tend to rank higher than the other groups.
The Kruskal Wallis test doesn’t involve medians or other distributional properties—just the ranks. In fact, by evaluating ranks, it rolls up both the location and shape parameters into a single evaluation of each group’s average rank.
When their average ranks are unequal, you know a group’s distribution tends to produce higher or lower values than the others. However, you don’t know enough to draw conclusions specifically about the distributions’ locations (e.g., the medians).
However, when you hold the distribution shapes constant, the Kruskal Wallis test does tell us about the median. That’s not a property of the procedure itself but logic. If several distributions have the same shape, but the average ranks are shifted higher and lower, their medians must differ. But we can only draw that conclusion about the medians when the distributions have the same shapes.
These three distributions have the same shape, but the red and green are shifted right to higher values. Wherever the median falls on the blue distribution, it’ll be in the corresponding position in the red and blue distributions. In this case, the analysis can assess the medians.
But, if the shapes aren’t similar, we don’t know whether the location, shape, or a combination of the two produced the statistically significant Kruskal Wallis test.
Like all statistical analyses, the Kruskal Wallis test has assumptions. Ensuring that your data meet these assumptions is crucial.
Violating these assumptions can lead to incorrect conclusions.
Consider using the Kruskal Wallis test in the following cases:
Learn more about the Normal Distribution .
If you have 3 – 9 groups and more than 15 observations per group or 10 – 12 groups and more than 20 observations per group, you might want to use one-way ANOVA even when you have nonnormal data. The central limit theorem causes the sampling distributions to converge on normality, making ANOVA a suitable choice.
One-way ANOVA has several advantages over the Kruskal Wallis test, including the following:
In short, use this nonparametric method when you’re specifically interested in the medians, have ordinal data, or can’t use one-way ANOVA because you have a small, nonnormal sample.
Like one-way ANOVA, the Kruskal Wallis test is an “omnibus” test. Omnibus tests can tell you that not all your groups are equal, but it doesn’t specify which pairs of groups are different.
Specifically, the Kruskal Wallis test evaluates the following hypotheses:
Again, if the distributions have similar shapes, you can replace “average ranks” with “medians.”
Imagine you’re studying five different diets and their impact on weight loss. The Kruskal Wallis test can confirm that at least two diets have different results. However, it won’t tell you exactly which pairs of diets have statistically significant differences.
So, how do we solve this problem? Enter post hoc tests. Perform these analyses after (i.e., post) an omnibus analysis to identify specific pairs of groups with statistically significant differences. A standard option includes Dunn’s multiple comparisons procedure. Other options include performing a series of pairwise Mann-Whitney U tests with a Bonferroni correction or the lesser-known but potent Conover-Iman method.
Learn about Post Hoc Tests for ANOVA .
Imagine you’re a healthcare administrator analyzing the median number of unoccupied beds in three hospitals. Download the CSV dataset: KruskalWallisTest .
For this Kruskal Wallis test, the p-value is 0.029, which is less than the typical significance level of 0.05. Consequently, we can reject the null hypothesis that all groups have the same average rank. At least one group has a different average rank than the others.
Furthermore, if the three hospital distributions have the same shape, we can conclude that the medians differ.
At this point, we might decide to use a post hoc test to compare pairs of hospitals.
May 20, 2024 at 2:07 pm
Sir kruskal walllis test is Two tailed or one tailed test??
May 20, 2024 at 3:55 pm
It’s a one-tailed test in the same sense that the F-test for one-way ANOVA is one-tailed.
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In this video, we will explore the concept of hypothesis testing in statistics. Hypothesis testing is a fundamental method used to make inferences about populations based on sample data. This tutorial is perfect for students, professionals, or anyone interested in enhancing their statistical analysis skills.
Understanding hypothesis testing helps to:
1. Hypothesis:
2. Null Hypothesis (H0):
3. Alternative Hypothesis (H1):
4. Test Statistic:
5. P-value:
6. Significance Level (α):
7. Type I Error:
8. Type II Error:
1. Formulate Hypotheses:
2. Choose Significance Level:
3. Select Test Statistic:
4. Calculate Test Statistic:
5. Determine P-value:
6. Make a Decision:
Example 1: One-Sample t-test
Formulate Hypotheses:
Choose Significance Level:
Select Test Statistic:
Calculate Test Statistic:
Determine P-value:
Make a Decision:
Scientific Research:
Business and Economics:
Medicine and Healthcare:
For more detailed information and a comprehensive guide on understanding hypothesis testing, check out the full article on GeeksforGeeks: https://www.geeksforgeeks.org/understanding-hypothesis-testing/ . This article provides in-depth explanations, examples, and further readings to help you master this statistical technique.
By the end of this video, you’ll have a solid understanding of hypothesis testing, enabling you to conduct and interpret statistical analyses effectively.
Read the full article for more details: https://www.geeksforgeeks.org/understanding-hypothesis-testing/ .
Thank you for watching!
The blood microbiome is probably not real.
Up until recently, if bacteria were detected in your blood you would be in a world of trouble. Blood was long considered to be sterile, meaning free of viable microorganisms like bacteria. When disease-causing bacteria spread to the blood, they can cause a life-threatening septic shock.
But the use of DNA sequencing technology has allowed researchers to more easily detect something that had been reported as early as the late 1960s: bacteria can be found in the blood and not cause disease.
As we begin to map out and understand the complex microbial ecosystem that lives in our gut and elsewhere in the body, we contemplate an important question: is there such a thing as a blood microbiome?
Our large intestine is not sterile; it is teeming with bacteria. But there are parts of the body that were long thought to be devoid of microorganisms. The brain. Bones. A variety of internal fluids, like our synovial fluid and peritoneal fluid. And, importantly, the blood.
Blood is made up of a liquid called plasma filled with red blood cells, whose main function is to carry oxygen to our cells. It also transports white blood cells, important to monitor for and fight off infections, as well as platelets, involved in clotting.
In the 1960s, a team of Italian researchers published multiple papers describing “mycoplasm-like forms”—meaning shapes that look like a particular type of bacteria that often contaminate cells cultured in the lab—in the blood of healthy people. This finding was confirmed in 1977 by a different team, which reported that four out of the 60 blood samples they had drawn from healthy volunteers showed bacteria growing in them. These types of tests, however, were rudimentary compared to what we have access to now. In the 2000s, they were mostly supplanted by DNA testing.
While we can sequence the entire DNA of any bacteria found in the blood, the technique most often used is 16S rRNA gene sequencing. I have always admired physicists’ penchant for quirky names: gluons, neutrinos, and charm quarks. Molecular biologists, by comparison, tend to be more sober. Yes, we have genes like Sonic hedgehog and proteins called scramblases; usually, though, we have to contend with the dryness of “16S rRNA.” You see, RNA is a molecule with many uses. Messenger RNA (or mRNA) acts as a disposable copy of a gene, a template for the production of a specific protein. Transfer RNA (or tRNA) actually brings the building blocks of a protein to where they are being assembled. And ribosomal RNA (or rRNA) is the main component of the giant protein factories in our cells known as ribosomes. One of its subunits is made up of, among others, a particular string of RNA known as the 16S rRNA.
The cool thing about the gene that codes for this 16S rRNA molecule is that it is very old and it mutates at a slow rate. By reading its precise sequence, scientists can tell which species it belongs to. Most of the studies of the putative blood microbiome use this technique to tell which species of bacteria are present in the blood being tested. The limitation of this test, however, is that dead bacteria have DNA too. The fact that DNA from the 16S rRNA gene of a precise bacterial species was detected in someone’s blood does not mean these bacteria were alive. For there to be a microbiome in the blood, these microorganisms need to live.
Which brings us to another important point of discussion. In order for scientists to agree that a blood microbiome exists, they first need to decide on the definition of a microbiome, and this is still a point of contention. In 2020, while companies were more than happy to sell hyped-up services testing your gut microbiome and claiming to interpret what it meant for your health, actual experts in the field met to agree on just what the word meant. “We are lacking,” they wrote , “a clear commonly agreed definition of the term ‘microbiome’.” For example, do viruses qualify? A microbiome implies life but viruses live on the edge, pun intended: they have the genetic blueprint for life yet they cannot reproduce on their own.
These experts proposed that the word “microbiome” should refer to the sum of two things: the microbiota, meaning the living microorganisms themselves, and their theatre of activity. It’s like saying that the Earth is not simply the life forms it houses, but also all of their individual components, and the traces they leave behind, and the environmental conditions in which they thrive or die. The microbiome is made up of bacteria and other microorganisms, yes, but also their proteins, lipids, sugars, and DNA and RNA molecules, as well as the signalling molecules and toxins that get exchanged within their theatre. (This is where viruses were sorted, by the way: not as part of the living microbiota but as belonging to the theatre of activity of the microbiome.)
The microbiome is a community, and this community has a distinct habitat.
So, what does the evidence say? Is our blood truly host to a thriving community of microorganisms or is something else going on?
Initial studies of the alleged blood microbiome were small . The amounts of bacteria that were being reported based on DNA sequencing were tiny. If this microbiome existed, it seemed sparse, more “asteroid field in real life” than “asteroid field in the movies.”
An issue looming over this early research is that of contamination. If bacteria are detected in a blood sample, were they really in the blood… or did they contaminate supplies along the way? When blood is drawn, the skin, which has its own microbiome, is punctured. The area is usually swabbed with alcohol to kill bacteria, and the supplies used should be sterile, but suffice to say that from the blood draw to the DNA extraction to the DNA amplification to the sequencing of this DNA, bacteria can be introduced into the system. In fact, it is such common knowledge that certain bacteria are found inside of the laboratory kits used by scientists that this ecosystem has its own name: the kitome. One way to rule out these contaminants is to simultaneously run negative controls alongside samples every step of the way, to make sure that these negative controls are indeed free of bacteria. But early papers rarely reported when controls were used.
Last year, results from what purports to be the largest study ever into the question of whether the blood microbiome exists were published in Nature Microbiology . A total of 9,770 healthy individuals were tested. The conclusion? Yes, some bacteria could be found in their blood, but the evidence contradicted the claim of an ecosystem. In 84% of the samples tested, no bacteria were detected. In most of the other samples, only one species was found. In an ecosystem, you would expect to see species appearing together repeatedly, but this was not the case here. And the species they found most often in their samples were known to contaminate these types of laboratory experiments.
So, what were the few bacteria found in the blood and not recognized as contaminants doing there in the first place if they were not part of a healthy microbiome? The authors lean toward an alternative explanation that had been floated for many years: these bacteria are transient. They end up in the blood from other parts of the body, either because of some minor leak or through their active transportation into the blood by agents such as dendritic cells. Like pedestrians wandering off onto the highway, these bacteria do not normally live in the blood but they can be seen there when we look at the right moment.
This blood microbiome story could end here and simply be an interesting example of scientific research homing in on a curious finding, testing a hypothesis, and ultimately refuting it (or at the very least providing strong evidence against it). But given the incentives of modern research and the social-media spotlight cast on the academic literature, there are two slightly worrying angles here that merit discussion.
Scientists are more and more incentivized to find practical applications for their research. It’s not enough, for example, to study bacteria that survive at incredibly high temperatures; we must be assured that the DNA replication enzyme these bacteria possess will one day be used in laboratories all over the world to conduct research, identify criminals, and test samples for the presence of a pandemic-causing coronavirus.
In researching this topic, I came across many papers claiming the existence of “blood microbiome signatures” for certain diseases that are not known to be infectious. We are thus not talking about infections leaking in the blood and causing sepsis. I saw reports of signatures for cardiovascular disease , liver disease , heart attacks , even for gastrointestinal disease in dogs . The idea is that these signatures could soon be turned into (profitable) diagnostic tests. The problem, of course, is that these studies are based on the hypothesis that a blood microbiome is real; that its equilibrium can be affected by disease; and that these changes can be reliably detected and interpreted.
But if the blood microbiome is imaginary, we are just chasing ghosts. This is not unlike the time that scientists were publishing signatures of microRNAs in the blood for every possible cancer. When I looked at the published literature in grad school, I realized that the multiple signatures reported for a single cancer barely overlapped . They were just chance findings. Compare enough variables in a small sample set and you will find what appears to be an association.
My second concern is that the transitory leakage of bacteria into the blood, as evidenced by the recent Nature Microbiology paper, will be used as confirmation of a pseudoscientific entity: leaky gut syndrome. At the end of their paper, the researchers hypothesize that these bacteria end up in the blood because the integrity of certain barriers in the body are compromised during disease or during periods of stress. The “net” in our gut gets a bit porous, and some of our colon’s bacteria end up in circulation, though they are not causing disease as far as we can tell. A form of leaky gut is known to exist in certain intestinal diseases , likely to be a consequence and not a cause. But leaky gut syndrome, favoured by non-evidence-based practitioners, does not appear to be real, yet many websites portray it as the one true cause of all diseases, a real epidemic. Nuanced scientific findings have a history of being stolen, distorted, and toyed with by fake doctors to give credence to their pet theories. Though I have yet to see examples of it, I suspect work done on this hypothesized blood microbiome will similarly get weaponized.
You have been warned.
Take-home message: - Our blood was long considered to be sterile, meaning free of viable microbes, unless a dangerous infection leaked into it, causing sepsis - Studies have provided evidence for the presence of bacteria in the blood of some healthy humans, leading to the hypothesis that, much like in our gut, our blood is host to a microbiome - The largest study ever done on the topic provided strong evidence against this hypothesis. It seems that when non-disease-causing bacteria find themselves in our blood, it is temporary and occasional
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The story linking nutrition and health has unexpected twists 28 jun 2024.
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COMMENTS
Hypothesis testing example. You want to test whether there is a relationship between gender and height. Based on your knowledge of human physiology, you formulate a hypothesis that men are, on average, taller than women. To test this hypothesis, you restate it as: H 0: Men are, on average, not taller than women. H a: Men are, on average, taller ...
If the biologist set her significance level \(\alpha\) at 0.05 and used the critical value approach to conduct her hypothesis test, she would reject the null hypothesis if her test statistic t* were less than -1.6939 (determined using statistical software or a t-table):s-3-3. Since the biologist's test statistic, t* = -4.60, is less than -1.6939, the biologist rejects the null hypothesis.
Hypothesis Testing Example. Researchers want to determine if a new educational program improves student performance on standardized tests. They randomly assign 30 students to a control group, which follows the standard curriculum, and another 30 students to a treatment group, which participates in the new educational program. After a semester ...
Example 1: Biology. Hypothesis tests are often used in biology to determine whether some new treatment, fertilizer, pesticide, chemical, etc. causes increased growth, stamina, immunity, etc. in plants or animals. For example, suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than ...
Test Statistic: z = x¯¯¯ −μo σ/ n−−√ z = x ¯ − μ o σ / n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4 7.1. 4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.
Developing a hypothesis (with example) 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. Example: Research question.
Unit 12: Significance tests (hypothesis testing) Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values ...
In hypothesis testing, we either reject or accept the null hypothesis. In our example, die 1 and die 2 are null and alternate hypotheses respectively. If you think about it intuitively, if the die lands on 1 or 2, it's more likely die 2 because it has more probability to land on 1 or 2.
The null hypothesis, denoted as H 0, is the hypothesis that the sample data occurs purely from chance. The alternative hypothesis, denoted as H 1 or H a, is the hypothesis that the sample data is influenced by some non-random cause. Hypothesis Tests. A hypothesis test consists of five steps: 1. State the hypotheses. State the null and ...
Hypothesis testing is a method of statistical inference that considers the null hypothesis H ₀ vs. the alternative hypothesis H a, where we are typically looking to assess evidence against H ₀. Such a test is used to compare data sets against one another, or compare a data set against some external standard. The former being a two sample ...
Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant. It involves the setting up of a null hypothesis and an alternate hypothesis. There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
It tests the null hypothesis that the population variances are equal (called homogeneity of variance or homoscedasticity). Suppose the resulting p-value of Levene's test is less than the significance level (typically 0.05).In that case, the obtained differences in sample variances are unlikely to have occurred based on random sampling from a population with equal variances.
Hypothesis testing is a crucial procedure to perform when you want to make inferences about a population using a random sample. These inferences include estimating population properties such as the mean, differences between means, proportions, and the relationships between variables. This post provides an overview of statistical hypothesis testing.
In this video there was no critical value set for this experiment. In the last seconds of the video, Sal briefly mentions a p-value of 5% (0.05), which would have a critical of value of z = (+/-) 1.96. Since the experiment produced a z-score of 3, which is more extreme than 1.96, we reject the null hypothesis.
Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence.
Example 8.4.7. Joon believes that 50% of first-time brides in the United States are younger than their grooms. She performs a hypothesis test to determine if the percentage is the same or different from 50%. Joon samples 100 first-time brides and 53 reply that they are younger than their grooms.
Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. The null and alternative hypotheses are: H0: x¯ = 4.5,Ha: x¯ > 4.5 H 0: x ¯ = 4.5, H a: x ¯ > 4.5.
ANOVA and MANOVA tests are used when comparing the means of more than two groups (e.g., the average heights of children, teenagers, and adults). Predictor variable. Outcome variable. Research question example. Paired t-test. Categorical. 1 predictor. Quantitative. groups come from the same population.
Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used ...
Full Hypothesis Test Examples. Example 8.6.4 8.6. 4. Statistics students believe that the mean score on the first statistics test is 65. A statistics instructor thinks the mean score is higher than 65. He samples ten statistics students and obtains the scores 65 65 70 67 66 63 63 68 72 71.
Hypothesis Testing Examples and Solutions. Let's delve into some common examples of hypothesis testing and provide solutions or interpretations for each scenario. Example: Testing the Mean. Scenario: A coffee shop owner believes that the average waiting time for customers during peak hours is 5 minutes. To test this, the owner takes a random ...
Oct 3, 2023. 77. Introduction: Hypothesis testing is a fundamental concept in statistics that helps us make data-driven decisions in various fields, from business to scientific research. In this ...
View Solution to Question 1. Question 2. A professor wants to know if her introductory statistics class has a good grasp of basic math. Six students are chosen at random from the class and given a math proficiency test. The professor wants the class to be able to score above 70 on the test. The six students get the following scores:62, 92, 75 ...
The Kruskal Wallis test is a nonparametric hypothesis test that compares three or more independent groups. Statisticians also refer to it as one-way ANOVA on ranks. ... The procedure ranks all the sample data from low to high. Then it averages the ranks for all groups. If the results are statistically significant, the average group ranks are ...
Hyp Mean - One Sample (data) One Sample Hypothesis Test for the Mean PROBLEM 1 1100 Ho: Mean SAT =1100 Level of Significance 0.05 Ha: Mean SAT greater than 1100 Sample Size 95 Upper-tailed test Sample Mean 1293.16 With a p-value of .0000 < .05, reject the null hypothesis Standard Deviation 121.28 Conclude that the mean SAT is less than 1100.
Understanding Hypothesis Testing. In this video, we will explore the concept of hypothesis testing in statistics. Hypothesis testing is a fundamental method used to make inferences about populations based on sample data. This tutorial is perfect for students, professionals, or anyone interested in enhancing their statistical analysis skills.
Putting the diagnostic cart before the horse This blood microbiome story could end here and simply be an interesting example of scientific research homing in on a curious finding, testing a hypothesis, and ultimately refuting it (or at the very least providing strong evidence against it).