Chapter 9.6, Problem 9.42E

Independent random samples of $n_1=140$ and $n_2=140$ observations were randomly selected from binomial populations 1 and 2, respectively. Sample 1 had 74 successes, and sample 2 had 81 successes.

a. Suppose you have no preconceived idea as to which parameter, $p_1$ or $p_2$ , is the larger, but you want only to detect a difference between the two parameters if one exists. What should you choose as the alternative hypothesis for a statistical test? The null hypothesis?

b. Calculate the standard error of the difference in the two sample proportions, $\left({\hat{p}}_1-{\hat{p}}_2\right)$ . Make sure to use the pooled estimate for the common value of p.

c. Calculate the test statistic that you would use for the test in part a. Based on your knowledge of the standard normal distribution, is this a likely or unlikely observation, assuming that $H_0$ is true and the two population proportions are the same?

d. p-value approach: Find the p-value for the test. Test for a significant difference in the population proportions at the 1% significance level.

e. Critical value approach: Find the rejection region when $\alpha =.01$ . Do the data provide sufficient evidence to indicate a difference in the population proportions?