Chapter 8.3, Problem 25E

The article "Unmarried Couples More Likely to Be Interracial" (San Luis Obispo Tribune, March 13, 2002) reported that 7% of married couples in the United States are mixed racially or ethnically. Consider the population consisting of all married couples in the United States. (Hint: See Example 8.9.)

1. a. A random sample of n = 100 couples will be selected from this population and $\hat{p}$, the proportion of couples that are mixed racially or ethnically, will be computed. What are the mean and standard deviation of the sampling distribution of $\hat{p}$?
2. b. Is it reasonable to assume that the sampling distribution of $\hat{p}$ is approximately normal for random samples of size n = 100? Explain.
3. c. Suppose that the sample size is n = 200 rather than n = 100, as in Part (b). Does the change in sample size change the mean and standard deviation of the sampling distribution of $\hat{p}$ ? If so, what are the new values for the mean and standard deviation? If not, explain why not.
4. d. Is it reasonable to assume that the sampling distribution of $\hat{p}$ is approximately normal for random samples of size n = 200? Explain.
5. e. When n = 200, what is the probability that the proportion of couples in the sample who are racially or ethnically mixed will be greater than . 10?