The General Social Survey asked 1373 men and 993 women in the United States whether they agreed that they were generally optimistic about the future. The results are presented in the following table.

a. Compute the value of the X2 statistic for testing the hypothesis that the opinion on optimism is independent of gender.

b. Compute the proportion of men who were optimistic.

c. Compute the proportion of women who were optimistic.

d. Compute the test statistic z for testing the null hypothesis that the two proportions are equal versus the alternative that they are not equal.

e. Show that X2 = z2.

f. Use technology to find the P-value for each test. Show that the P-values are equal.
g. Conclude that when a contingency table has two rows and two columns, the chi-square test is equivalent to the test for the difference between proportions.

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Male Female Optimistic 1148 Pessimistic 815 225 178