The obvious answer to your question is that it makes sense to apply an unpooled t-test when: the above assumptions are met; you have good reason such as a scientific model to believe that the population variances are the same; and the number of samples is very small so that there would be a noteworthy difference between the tests.
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Active 3 years, 8 months ago. Viewed 11k times. Should we always check if the variance of the two normal distributions is the same? Improve this question. Grace Mahoney 1 1 gold badge 1 1 silver badge 8 8 bronze badges. Add a comment. Active Oldest Votes. Improve this answer.
Impact on significance level is what most people are concerned about when they consider that there might be different population variances; that's specifically the issue that the Welch test is attempting to fix. The counterexamples: If you don't have much time, have a glance on this post by Daniel Lakens. Adam Przedniczek Adam Przedniczek 1, 2 2 gold badges 10 10 silver badges 24 24 bronze badges.
The test that assumes equal population variances is referred to as the pooled t-test. Pooling refers to finding a weighted average of the two independent sample variances. The pooled test statistic uses a weighted average of the two sample variances. The hypothesis test procedure will follow the same steps as the previous section. It may be difficult to verify that two population variances might be equal based on sample data. The F-test is commonly used to test variances but is not robust.
Small departures from normality greatly impact the outcome making the results of the F-test unreliable. It can be difficult to decide if a significant outcome from an F-test is due to the differences in variances or non-normality.
Growth of pine seedlings in two different substrates was measured. The normality assumption is more important when the two groups have small sample sizes than for larger sample sizes. Normal distributions do not have extreme values, or outliers.
You can check these two features of a normal distribution with graphs. 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. 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. 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.
For the figure above, with the F test statistic of 1. 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.
Because our test statistic of 2. 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. The software shows results for a two-sided test and for one-sided tests. 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. 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 The software shows a p- value of 0. 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. 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.
The Two-Sample t -Test. What is the two-sample t -test? When can I use the test? What if I have more than two groups? What if the variances for my two groups are not equal?
Using the two-sample t -test The sections below discuss what is needed to perform the test, checking our data, how to perform the test and statistical details. What do we need? Here are a couple of examples: We have students who speak English as their first language and students who do not.
All students take a reading test. Our two groups are the native English speakers and the non-native speakers. Our measurements are the test scores.
Our idea is that the mean test scores for the underlying populations of native and non-native English speakers are not the same. We want to know if the mean score for the population of native English speakers is different from the people who learned English as a second language. We measure the grams of protein in two different brands of energy bars.
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