Men
Women
You can clearly see some overlap in the body fat measurements for the men and women in our sample, but also some differences. Just by looking at the data, it's hard to draw any solid conclusions about whether the underlying populations of men and women at the gym have the same mean body fat. That is the value of statistical tests – they provide a common, statistically valid way to make decisions, so that everyone makes the same decision on the same set of data values.
Let’s start by answering: Is the two-sample t -test an appropriate method to evaluate the difference in body fat between men and women?
Before jumping into analysis, we should always take a quick look at the data. The figure below shows histograms and summary statistics for the men and women.
The two histograms are on the same scale. From a quick look, we can see that there are no very unusual points, or outliers . The data look roughly bell-shaped, so our initial idea of a normal distribution seems reasonable.
Examining the summary statistics, we see that the standard deviations are similar. This supports the idea of equal variances. We can also check this using a test for variances.
Based on these observations, the two-sample t -test appears to be an appropriate method to test for a difference in means.
For each group, we need the average, standard deviation and sample size. These are shown in the table below.
Women | 10 | 22.29 | 5.32 |
Men | 13 | 14.95 | 6.84 |
Without doing any testing, we can see that the averages for men and women in our samples are not the same. But how different are they? Are the averages “close enough” for us to conclude that mean body fat is the same for the larger population of men and women at the gym? Or are the averages too different for us to make this conclusion?
We'll further explain the principles underlying the two sample t -test in the statistical details section below, but let's first proceed through the steps from beginning to end. We start by calculating our test statistic. This calculation begins with finding the difference between the two averages:
$ 22.29 - 14.95 = 7.34 $
This difference in our samples estimates the difference between the population means for the two groups.
Next, we calculate the pooled standard deviation. This builds a combined estimate of the overall standard deviation. The estimate adjusts for different group sizes. First, we calculate the pooled variance:
$ s_p^2 = \frac{((n_1 - 1)s_1^2) + ((n_2 - 1)s_2^2)} {n_1 + n_2 - 2} $
$ s_p^2 = \frac{((10 - 1)5.32^2) + ((13 - 1)6.84^2)}{(10 + 13 - 2)} $
$ = \frac{(9\times28.30) + (12\times46.82)}{21} $
$ = \frac{(254.7 + 561.85)}{21} $
$ =\frac{816.55}{21} = 38.88 $
Next, we take the square root of the pooled variance to get the pooled standard deviation. This is:
$ \sqrt{38.88} = 6.24 $
We now have all the pieces for our test statistic. We have the difference of the averages, the pooled standard deviation and the sample sizes. We calculate our test statistic as follows:
$ t = \frac{\text{difference of group averages}}{\text{standard error of difference}} = \frac{7.34}{(6.24\times \sqrt{(1/10 + 1/13)})} = \frac{7.34}{2.62} = 2.80 $
To evaluate the difference between the means in order to make a decision about our gym programs, we compare the test statistic to a theoretical value from the t- distribution. This activity involves four steps:
Let’s look at the body fat data and the two-sample t -test using statistical terms.
Our null hypothesis is that the underlying population means are the same. The null hypothesis is written as:
$ H_o: \mathrm{\mu_1} =\mathrm{\mu_2} $
The alternative hypothesis is that the means are not equal. This is written as:
$ H_o: \mathrm{\mu_1} \neq \mathrm{\mu_2} $
We calculate the average for each group, and then calculate the difference between the two averages. This is written as:
$\overline{x_1} - \overline{x_2} $
We calculate the pooled standard deviation. This assumes that the underlying population variances are equal. The pooled variance formula is written as:
The formula shows the sample size for the first group as n 1 and the second group as n 2 . The standard deviations for the two groups are s 1 and s 2 . This estimate allows the two groups to have different numbers of observations. The pooled standard deviation is the square root of the variance and is written as s p .
What if your sample sizes for the two groups are the same? In this situation, the pooled estimate of variance is simply the average of the variances for the two groups:
$ s_p^2 = \frac{(s_1^2 + s_2^2)}{2} $
The test statistic is calculated as:
$ t = \frac{(\overline{x_1} -\overline{x_2})}{s_p\sqrt{1/n_1 + 1/n_2}} $
The numerator of the test statistic is the difference between the two group averages. It estimates the difference between the two unknown population means. The denominator is an estimate of the standard error of the difference between the two unknown population means.
Technical Detail: For a single mean, the standard error is $ s/\sqrt{n} $ . The formula above extends this idea to two groups that use a pooled estimate for s (standard deviation), and that can have different group sizes.
We then compare the test statistic to a t value with our chosen alpha value and the degrees of freedom for our data. Using the body fat data as an example, we set α = 0.05. The degrees of freedom ( df ) are based on the group sizes and are calculated as:
$ df = n_1 + n_2 - 2 = 10 + 13 - 2 = 21 $
The formula shows the sample size for the first group as n 1 and the second group as n 2 . Statisticians write the t value with α = 0.05 and 21 degrees of freedom as:
$ t_{0.05,21} $
The t value with α = 0.05 and 21 degrees of freedom is 2.080. There are two possible results from our comparison:
When the variances for the two groups are not equal, we cannot use the pooled estimate of standard deviation. Instead, we take the standard error for each group separately. The test statistic is:
$ t = \frac{ (\overline{x_1} - \overline{x_2})}{\sqrt{s_1^2/n_1 + s_2^2/n_2}} $
The numerator of the test statistic is the same. It is the difference between the averages of the two groups. The denominator is an estimate of the overall standard error of the difference between means. It is based on the separate standard error for each group.
The degrees of freedom calculation for the t value is more complex with unequal variances than equal variances and is usually left up to statistical software packages. The key point to remember is that if you cannot use the pooled estimate of standard deviation, then you cannot use the simple formula for the degrees of freedom.
The normality assumption is more important when the two groups have small sample sizes than for larger sample sizes.
Normal distributions are symmetric, which means they are “even” on both sides of the center. Normal distributions do not have extreme values, or outliers. You can check these two features of a normal distribution with graphs. Earlier, we decided that the body fat data was “close enough” to normal to go ahead with the assumption of normality. The figure below shows a normal quantile plot for men and women, and supports our decision.
You can also perform a formal test for normality using software. The figure above shows results of testing for normality with JMP software. We test each group separately. Both the test for men and the test for women show that we cannot reject the hypothesis of a normal distribution. We can go ahead with the assumption that the body fat data for men and for women are normally distributed.
Testing for unequal variances is complex. We won’t show the calculations in detail, but will show the results from JMP software. The figure below shows results of a test for unequal variances for the body fat data.
Without diving into details of the different types of tests for unequal variances, we will use the F test. Before testing, we decide to accept a 10% risk of concluding the variances are equal when they are not. This means we have set α = 0.10.
Like most statistical software, JMP shows the p -value for a test. This is the likelihood of finding a more extreme value for the test statistic than the one observed. It’s difficult to calculate by hand. For the figure above, with the F test statistic of 1.654, the p- value is 0.4561. This is larger than our α value: 0.4561 > 0.10. We fail to reject the hypothesis of equal variances. In practical terms, we can go ahead with the two-sample t -test with the assumption of equal variances for the two groups.
Using a visual, you can check to see if your test statistic is a more extreme value in the distribution. The figure below shows a t- distribution with 21 degrees of freedom.
Since our test is two-sided and we have set α = .05, the figure shows that the value of 2.080 “cuts off” 2.5% of the data in each of the two tails. Only 5% of the data overall is further out in the tails than 2.080. Because our test statistic of 2.80 is beyond the cut-off point, we reject the null hypothesis of equal means.
The figure below shows results for the two-sample t -test for the body fat data from JMP software.
The results for the two-sample t -test that assumes equal variances are the same as our calculations earlier. The test statistic is 2.79996. The software shows results for a two-sided test and for one-sided tests. The two-sided test is what we want (Prob > |t|). Our null hypothesis is that the mean body fat for men and women is equal. Our alternative hypothesis is that the mean body fat is not equal. The one-sided tests are for one-sided alternative hypotheses – for example, for a null hypothesis that mean body fat for men is less than that for women.
We can reject the hypothesis of equal mean body fat for the two groups and conclude that we have evidence body fat differs in the population between men and women. The software shows a p -value of 0.0107. We decided on a 5% risk of concluding the mean body fat for men and women are different, when they are not. It is important to make this decision before doing the statistical test.
The figure also shows the results for the t- test that does not assume equal variances. This test does not use the pooled estimate of the standard deviation. As was mentioned above, this test also has a complex formula for degrees of freedom. You can see that the degrees of freedom are 20.9888. The software shows a p- value of 0.0086. Again, with our decision of a 5% risk, we can reject the null hypothesis of equal mean body fat for men and women.
If you have more than two independent groups, you cannot use the two-sample t- test. You should use a multiple comparison method. ANOVA, or analysis of variance, is one such method. Other multiple comparison methods include the Tukey-Kramer test of all pairwise differences, analysis of means (ANOM) to compare group means to the overall mean or Dunnett’s test to compare each group mean to a control mean.
If your sample size is very small, it might be hard to test for normality. In this situation, you might need to use your understanding of the measurements. For example, for the body fat data, the trainer knows that the underlying distribution of body fat is normally distributed. Even for a very small sample, the trainer would likely go ahead with the t -test and assume normality.
What if you know the underlying measurements are not normally distributed? Or what if your sample size is large and the test for normality is rejected? In this situation, you can use nonparametric analyses. These types of analyses do not depend on an assumption that the data values are from a specific distribution. For the two-sample t -test, the Wilcoxon rank sum test is a nonparametric test that could be used.
Introduction.
The independent t-test, also called the two sample t-test, independent-samples t-test or student's t-test, is an inferential statistical test that determines whether there is a statistically significant difference between the means in two unrelated groups.
The null hypothesis for the independent t-test is that the population means from the two unrelated groups are equal:
H 0 : u 1 = u 2
In most cases, we are looking to see if we can show that we can reject the null hypothesis and accept the alternative hypothesis, which is that the population means are not equal:
H A : u 1 ≠ u 2
To do this, we need to set a significance level (also called alpha) that allows us to either reject or accept the alternative hypothesis. Most commonly, this value is set at 0.05.
In order to run an independent t-test, you need the following:
Unrelated groups, also called unpaired groups or independent groups, are groups in which the cases (e.g., participants) in each group are different. Often we are investigating differences in individuals, which means that when comparing two groups, an individual in one group cannot also be a member of the other group and vice versa. An example would be gender - an individual would have to be classified as either male or female – not both.
The independent t-test requires that the dependent variable is approximately normally distributed within each group.
Note: Technically, it is the residuals that need to be normally distributed, but for an independent t-test, both will give you the same result.
You can test for this using a number of different tests, but the Shapiro-Wilks test of normality or a graphical method, such as a Q-Q Plot, are very common. You can run these tests using SPSS Statistics, the procedure for which can be found in our Testing for Normality guide. However, the t-test is described as a robust test with respect to the assumption of normality. This means that some deviation away from normality does not have a large influence on Type I error rates. The exception to this is if the ratio of the smallest to largest group size is greater than 1.5 (largest compared to smallest).
If you find that either one or both of your group's data is not approximately normally distributed and groups sizes differ greatly, you have two options: (1) transform your data so that the data becomes normally distributed (to do this in SPSS Statistics see our guide on Transforming Data ), or (2) run the Mann-Whitney U test which is a non-parametric test that does not require the assumption of normality (to run this test in SPSS Statistics see our guide on the Mann-Whitney U Test ).
The independent t-test assumes the variances of the two groups you are measuring are equal in the population. If your variances are unequal, this can affect the Type I error rate. The assumption of homogeneity of variance can be tested using Levene's Test of Equality of Variances, which is produced in SPSS Statistics when running the independent t-test procedure. If you have run Levene's Test of Equality of Variances in SPSS Statistics, you will get a result similar to that below:
This test for homogeneity of variance provides an F -statistic and a significance value ( p -value). We are primarily concerned with the significance value – if it is greater than 0.05 (i.e., p > .05), our group variances can be treated as equal. However, if p < 0.05, we have unequal variances and we have violated the assumption of homogeneity of variances.
If the Levene's Test for Equality of Variances is statistically significant, which indicates that the group variances are unequal in the population, you can correct for this violation by not using the pooled estimate for the error term for the t -statistic, but instead using an adjustment to the degrees of freedom using the Welch-Satterthwaite method. In all reality, you will probably never have heard of these adjustments because SPSS Statistics hides this information and simply labels the two options as "Equal variances assumed" and "Equal variances not assumed" without explicitly stating the underlying tests used. However, you can see the evidence of these tests as below:
From the result of Levene's Test for Equality of Variances, we can reject the null hypothesis that there is no difference in the variances between the groups and accept the alternative hypothesis that there is a statistically significant difference in the variances between groups. The effect of not being able to assume equal variances is evident in the final column of the above figure where we see a reduction in the value of the t -statistic and a large reduction in the degrees of freedom (df). This has the effect of increasing the p -value above the critical significance level of 0.05. In this case, we therefore do not accept the alternative hypothesis and accept that there are no statistically significant differences between means. This would not have been our conclusion had we not tested for homogeneity of variances.
When reporting the result of an independent t-test, you need to include the t -statistic value, the degrees of freedom (df) and the significance value of the test ( p -value). The format of the test result is: t (df) = t -statistic, p = significance value. Therefore, for the example above, you could report the result as t (7.001) = 2.233, p = 0.061.
In order to provide enough information for readers to fully understand the results when you have run an independent t-test, you should include the result of normality tests, Levene's Equality of Variances test, the two group means and standard deviations, the actual t-test result and the direction of the difference (if any). In addition, you might also wish to include the difference between the groups along with a 95% confidence interval. For example:
Inspection of Q-Q Plots revealed that cholesterol concentration was normally distributed for both groups and that there was homogeneity of variance as assessed by Levene's Test for Equality of Variances. Therefore, an independent t-test was run on the data with a 95% confidence interval (CI) for the mean difference. It was found that after the two interventions, cholesterol concentrations in the dietary group (6.15 ± 0.52 mmol/L) were significantly higher than the exercise group (5.80 ± 0.38 mmol/L) ( t (38) = 2.470, p = 0.018) with a difference of 0.35 (95% CI, 0.06 to 0.64) mmol/L.
To know how to run an independent t-test in SPSS Statistics, see our SPSS Statistics Independent-Samples T-Test guide. Alternatively, you can carry out an independent-samples t-test using Excel, R and RStudio .
Topics: Hypothesis Testing , Data Analysis
In statistics, t-tests are a type of hypothesis test that allows you to compare means. They are called t-tests because each t-test boils your sample data down to one number, the t-value. If you understand how t-tests calculate t-values, you’re well on your way to understanding how these tests work.
In this series of posts, I'm focusing on concepts rather than equations to show how t-tests work. However, this post includes two simple equations that I’ll work through using the analogy of a signal-to-noise ratio.
Minitab Statistical Software offers the 1-sample t-test, paired t-test, and the 2-sample t-test. Let's look at how each of these t-tests reduce your sample data down to the t-value.
Understanding this process is crucial to understanding how t-tests work. I'll show you the formula first, and then I’ll explain how it works.
Please notice that the formula is a ratio. A common analogy is that the t-value is the signal-to-noise ratio.
The numerator is the signal. You simply take the sample mean and subtract the null hypothesis value. If your sample mean is 10 and the null hypothesis is 6, the difference, or signal, is 4.
If there is no difference between the sample mean and null value, the signal in the numerator, as well as the value of the entire ratio, equals zero. For instance, if your sample mean is 6 and the null value is 6, the difference is zero.
As the difference between the sample mean and the null hypothesis mean increases in either the positive or negative direction, the strength of the signal increases.
The denominator is the noise. The equation in the denominator is a measure of variability known as the standard error of the mean . This statistic indicates how accurately your sample estimates the mean of the population. A larger number indicates that your sample estimate is less precise because it has more random error.
This random error is the “noise.” When there is more noise, you expect to see larger differences between the sample mean and the null hypothesis value even when the null hypothesis is true . We include the noise factor in the denominator because we must determine whether the signal is large enough to stand out from it.
Both the signal and noise values are in the units of your data. If your signal is 6 and the noise is 2, your t-value is 3. This t-value indicates that the difference is 3 times the size of the standard error. However, if there is a difference of the same size but your data have more variability (6), your t-value is only 1. The signal is at the same scale as the noise.
In this manner, t-values allow you to see how distinguishable your signal is from the noise. Relatively large signals and low levels of noise produce larger t-values. If the signal does not stand out from the noise, it’s likely that the observed difference between the sample estimate and the null hypothesis value is due to random error in the sample rather than a true difference at the population level.
Many people are confused about when to use a paired t-test and how it works. I’ll let you in on a little secret. The paired t-test and the 1-sample t-test are actually the same test in disguise! As we saw above, a 1-sample t-test compares one sample mean to a null hypothesis value. A paired t-test simply calculates the difference between paired observations (e.g., before and after) and then performs a 1-sample t-test on the differences.
You can test this with this data set to see how all of the results are identical, including the mean difference, t-value, p-value, and confidence interval of the difference.
Understanding that the paired t-test simply performs a 1-sample t-test on the paired differences can really help you understand how the paired t-test works and when to use it. You just need to figure out whether it makes sense to calculate the difference between each pair of observations.
For example, let’s assume that “before” and “after” represent test scores, and there was an intervention in between them. If the before and after scores in each row of the example worksheet represent the same subject, it makes sense to calculate the difference between the scores in this fashion—the paired t-test is appropriate. However, if the scores in each row are for different subjects, it doesn’t make sense to calculate the difference. In this case, you’d need to use another test, such as the 2-sample t-test, which I discuss below.
Using the paired t-test simply saves you the step of having to calculate the differences before performing the t-test. You just need to be sure that the paired differences make sense!
When it is appropriate to use a paired t-test, it can be more powerful than a 2-sample t-test. For more information, go to Overview for paired t .
The 2-sample t-test takes your sample data from two groups and boils it down to the t-value. The process is very similar to the 1-sample t-test, and you can still use the analogy of the signal-to-noise ratio. Unlike the paired t-test, the 2-sample t-test requires independent groups for each sample.
The formula is below, and then some discussion.
For the 2-sample t-test, the numerator is again the signal, which is the difference between the means of the two samples. For example, if the mean of group 1 is 10, and the mean of group 2 is 4, the difference is 6.
The default null hypothesis for a 2-sample t-test is that the two groups are equal. You can see in the equation that when the two groups are equal, the difference (and the entire ratio) also equals zero. As the difference between the two groups grows in either a positive or negative direction, the signal becomes stronger.
In a 2-sample t-test, the denominator is still the noise, but Minitab can use two different values. You can either assume that the variability in both groups is equal or not equal, and Minitab uses the corresponding estimate of the variability. Either way, the principle remains the same: you are comparing your signal to the noise to see how much the signal stands out.
Just like with the 1-sample t-test, for any given difference in the numerator, as you increase the noise value in the denominator, the t-value becomes smaller. To determine that the groups are different, you need a t-value that is large.
Each type of t-test uses a procedure to boil all of your sample data down to one value, the t-value. The calculations compare your sample mean(s) to the null hypothesis and incorporates both the sample size and the variability in the data. A t-value of 0 indicates that the sample results exactly equal the null hypothesis. In statistics, we call the difference between the sample estimate and the null hypothesis the effect size. As this difference increases, the absolute value of the t-value increases.
That’s all nice, but what does a t-value of, say, 2 really mean? From the discussion above, we know that a t-value of 2 indicates that the observed difference is twice the size of the variability in your data. However, we use t-tests to evaluate hypotheses rather than just figuring out the signal-to-noise ratio. We want to determine whether the effect size is statistically significant.
To see how we get from t-values to assessing hypotheses and determining statistical significance, read the other post in this series, Understanding t-Tests: t-values and t-distributions .
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Course: ap®︎/college statistics > unit 11.
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Fortunately, a two sample t-test allows us to answer this question. Two Sample t-test: Formula. A two-sample t-test always uses the following null hypothesis: H 0: μ 1 = μ 2 (the two population means are equal) The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed:
Two-Sample T Test Hypotheses. Null hypothesis (H 0): Two population means are equal (µ 1 = µ 2). Alternative hypothesis (H A): Two population means are not equal (µ 1 ≠ µ 2). Again, when the p-value is less than or equal to your significance level, reject the null hypothesis. The difference between the two means is statistically significant.
Revised on June 22, 2023. A t test is a statistical test that is used to compare the means of two groups. It is often used in hypothesis testing to determine whether a process or treatment actually has an effect on the population of interest, or whether two groups are different from one another. t test example.
Fortunately, a two sample t-test allows us to answer this question. Two Sample t-test: Formula. A two-sample t-test always uses the following null hypothesis: H 0: μ 1 = μ 2 (the two population means are equal) The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed:
If that's below your significance level, then you would reject your null hypothesis and it would suggest the alternative that might be that, "Hey, maybe this mean "is greater than zero." On the other hand, a two-sample T test is where you're thinking about two different populations. For example, you could be thinking about a population of men ...
How Two-Sample T-tests Calculate T-Values. Use the 2-sample t-test when you want to analyze the difference between the means of two independent samples. This test is also known as the independent samples t-test. Click the link to learn more about its hypotheses, assumptions, and interpretations. Like the other t-tests, this procedure reduces ...
The results for the two-sample t-test that assumes equal variances are the same as our calculations earlier. The test statistic is 2.79996. The software shows results for a two-sided test and for one-sided tests. The two-sided test is what we want (Prob > |t|). Our null hypothesis is that the mean body fat for men and women is equal.
T-test definition, formula explanation, and assumptions. The T-test is the test, which allows us to analyze one or two sample means, depending on the type of t-test. Yes, the t-test has several types: One-sample t-test — compare the mean of one group against the specified mean generated from a population. For example, a manufacturer of mobile ...
Independent Samples T Tests Hypotheses. Independent samples t tests have the following hypotheses: Null hypothesis: The means for the two populations are equal. Alternative hypothesis: The means for the two populations are not equal.; If the p-value is less than your significance level (e.g., 0.05), you can reject the null hypothesis. The difference between the two means is statistically ...
The independent t-test, also called the two sample t-test, independent-samples t-test or student's t-test, is an inferential statistical test that determines whether there is a statistically significant difference between the means in two unrelated groups. ... The null hypothesis for the independent t-test is that the population means from the ...
The t-test for dependent samples is a statistical test for comparing the means from two dependent populations (or the difference between the means from two populations). The t-test is used when the differences are normally distributed. The samples also must be dependent. The formula for the t-test statistic is: t = D¯−μD (SD n√) t = D ...
And let's assume that we are working with a significance level of 0.05. So pause the video, and conduct the two sample T test here, to see whether there's evidence that the sizes of tomato plants differ between the fields. Alright, now let's work through this together. So like always, let's first construct our null hypothesis.
The 2-sample t-test takes your sample data from two groups and boils it down to the t-value. The process is very similar to the 1-sample t-test, and you can still use the analogy of the signal-to-noise ratio. Unlike the paired t-test, the 2-sample t-test requires independent groups for each sample. The formula is below, and then some discussion.
A one-sample t-test (to test the mean of a single group against a hypothesized mean); A two-sample t-test (to compare the means for two groups); or. A paired t-test (to check how the mean from the same group changes after some intervention). Decide on the alternative hypothesis: Two-tailed; Left-tailed; or. Right-tailed.
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/ap-statistics/xfb5d8e68:infere...
1. Two tailed test example: A factory uses two identical machines to produce plastic plates. You would expect both machines to produce the same number of plates per minute. Let μ1 = average number of plates produced by machine1 per minute. Let μ2 = average number of plates produced by machine2 per minute. We would expect μ1 to be equal to μ2.
First of all, if you have two groups, one testing one placebo, then it's 2 samples. If it is the same group before and after, then paired t-test. I'm trying to run a dependent sample t-test/paired sample t test through using data from a Qualtrics survey measuring two groups of people (one with social anxiety and one without on the effects of ...
00:37:48 - Create a two sample t-test and confidence interval with pooled variances (Example #4) 00:51:23 - Construct a two-sample t-test (Example #5) 00:59:47 - Matched Pair one sample t-test (Example #6) 01:09:38 - Use a match paired hypothesis test and provide a confidence interval for difference of means (Example #7) Practice ...
So an unusual value of t would be anything with absolute value greater than 2.04. Similarly, for 2 d.f. the bars are at +/- 4.3. The brilliance of the t-test is that if the null hypothesis is true then the two sample means and variances and the sample sizes are sufficient to compute the t-value and determine the associated p-value.
Student's t-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis.It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in ...
If this is not the case, you should instead use the Welch's t-test calculator. To perform a two sample t-test, simply fill in the information below and then click the "Calculate" button. Enter raw data Enter summary data. Sample 1. 301, 298, 295, 297, 304, 305, 309, 298, 291, 299, 293, 304. Sample 2.
A two sample t hypothesis tests also known as independent t-test is used to analyze the difference between two unknown population means. The Two-sample T-test is used when the two small samples (n< 30) are taken from two different populations and compared. The underlying chart makes use of the T distribution.
How t-Tests Work: t-Values, t-Distributions, and Probabilities. T-tests are statistical hypothesis tests that you use to analyze one or two sample means. Depending on the t-test that you use, you can compare a sample mean to a hypothesized value, the means of two independent samples, or the difference between paired samples.
Before we can actually conduct the hypothesis test, we'll have to derive the appropriate test statistic. Theorem. The test statistic for testing the difference in two population proportions, that is, for testing the null hypothesis \(H_0:p_1-p_2=0\) is: ... the proportion of "successes" in the two samples combined. Proof. Recall that: \(\hat{p ...
The populations from which the two samples are drawn are approximately normally distributed. The two populations are independent of each other. Unlike most other hypothesis tests in this book, the F test for equality of two variances is very sensitive to deviations from normality. If the two distributions are not normal, or close, the test can ...
The appropriate set of hypotheses for Alex's significance test is option D. What is Hypothesis Testing? Hypothesis testing is the testing of the relationship between two variables for something we needed to test. There is null hypothesis and alternate hypothesis. Null hypothesis, H₀ is the statement we need to test.