The t-tests are commonly used hypothesis tests. Researchers often find themselves in situations where they need to test to see if a sample comes from a certain population, and therefore test to see if the sample probably came from a population with that certain mean. Even more often, researchers will find themselves with two samples and want to know if the samples come from the same population, and will test to see if the samples probably come from populations with the same mean. Researchers also frequently find themselves asking if two sets of paired samples have equal means. In any case, the basic strategy is the same as for any hypothesis test. First, translate the question into null and alternative hypotheses, making sure that the null hypothesis includes an equal sign. Second, choose α. Third, compute the relevant statistics, here the t-score, from the sample or samples. Fourth, using the tables, decide if the sample statistic leads you to conclude that the sample came from a population where the null hypothesis is true or a population where the alternative is true.
The t-distribution is also used in testing hypotheses in other situations since there are other sampling distributions with the same t-distribution shape. So, remember how to use the t-tables for later chapters.
Statisticians have also found how to test to see if three or more samples come from populations with the same mean. That technique is known as "one-way analysis of variance". The approach used in analysis of variance is quite different from that used in the t-test. It will be covered in chapter, "The F-test and One-Way ANOVA".
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