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T-TESTS FOR INDEPENDENT SAMPLES T-tests are used to test whether the means of a continuous variable differ across two different groups. For example, do men and women differ in their levels of income, when measured as a continuous variable? Does crime vary between two parts of town? Do rich people live longer than poor people? Do high-performing students commit fewer acts of violence than do low-performing students? The t-test approach is shown graphically in Figure 12.1, which illustrates the incomes of men and women as boxplots (the lines in the middle of the boxes indicate the means rather than the medians).2 When the two groups are independent samples, the t-test is called the independent-samples t-test. Sometimes the continuous variable is called a “test variable” and the dichotomous variable is called a “grouping variable.” The t-test tests whether the difference of the means is significantly different from zero, that is, whether men and women have different incomes. The following hypotheses are posited: Key Point The independent-samples t-test is used when one variable is dichotomous and the other is continuous. H0: Men and women do not have different mean incomes (in the population). HA: Men and women do have different mean incomes (in the population). Alternatively, using the Greek letter m to refer to differences in the population, H0: μm = μf, and HA: μm ≠ μf. The formula for calculating the t-test test statistic (a tongue twister?) is As always, the computer calculates the test statistic and reports at what level it is significant. Such calculations are seldom done by hand. To further conceptual understanding of this formula, it is useful to relate it to the discussion of hypothesis testing in Chapter 10. First, note that the difference of means, appears in the numerator: the larger the difference of means, the larger the t-test test statistic, and the more likely we might reject the null hypothesis. Second, sp is the pooled variance of the two groups, that is, the weighted average of the variances of each group.3 Increases in the standard deviation decrease the test statistic. Thus, it is easier to reject the null hypotheses when two populations are clustered narrowly around their means than when they are spread widely around them. Finally, more observations (that is, increased information or larger n1 and n2) increase the size of the test statistic, making it easier to reject the null hypothesis. Figure 12.1 The T-Test: Mean Incomes by Gender
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Evan M. Berman (Essential Statistics for Public Managers and Policy Analysts)