The t-test is a statistical test for the difference between two means. It is typically used when the independent variable is dichotomous (e.g., sex), and the dependent variable is interval or ratio (e.g., income).
Consider a math test given to 1000 male and 1000 female 17-year-olds. The average for boys was 55.0 and for girls was 51.8. The SD for boys was 17.2 and for girls was 16.9. The difference in averages is 3.2. Is this just chance variation? The null hypothesis is no difference, so the test statistic is:
(3.2 - 0)/SE
(You might want to review the SD and the SE.)
But what is the SE of a difference? It is a kind of average of the SEs for each group. The SE for boys is 17.2/sqrt(1000) = .54. The SE for girls is 16.9/sqrt(1000) = 0.53. The SE for the difference is:
Sqrt(SE12 + SE22) = sqrt(.542 + .532) = .76
So the test statistic is:
Z = -3.2/0.76 = -4.2.
If we look up 4.2 on the normal table, we get p = 0.003.
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