Question: Mathematics and gender. On average, do males outperform females in mathematics? To answer this question, psychologists at the University of Minnesota compared the scores of male and female eighth-grade students who took a basic skills mathematics achievement test (American Educational Research Journal, Fall 1998). One form of the test consisted of 68 multiple-choice questions. A summary of the test scores is displayed in the table.

                                        Males                Females

Sample size                        1.764                  1.739

Mean                                  48.9                   48.4

Standard Deviation              12.96                   11.85

(a) Is there evidence of a difference between the true mean mathematics test scores of male and female eighth-graders?

(b) Use a 90% confidence interval to estimate the true difference in mean test scores between males and females. Does the confidence interval support the result of the test you conducted in part a?

(c) What assumptions about the distributions of the populations of test scores are necessary to ensure the validity of the inferences you made in parts a and b?

(d) What is the observed significance level of the test you conducted in part a?

(e) The researchers hypothesized that the distribution of test scores for males is more variable than the distribution for females. Test this claim at α = .05.