Chapter 8.3, Problem 25E

The article “Younger Adults More Likely than Their Elders to Prefer Reading the News” (October 6, 2016, pewresearch.org/fact-tank/2016/10/06/younger-adults-more-likely-than-their-elders-to-prefer-reading-news/, retrieved March 28, 2018) estimated that only 3% of those age 65 and older who prefer to watch the news, rather than to read or listen to news, watch the news online. This estimate was based on a large sample of adult Americans conducted by the Pew Research Center. Consider the population consisting of all adult Americans age 65 and older who prefer to watch the news and suppose that for this population the actual proportion who prefer to watch online is 0.03. (Hint: See Example 8.9.)

1. a. A random sample of n = 100 people will be selected from this population and $\hat{p}$, the proportion of people who prefer to watch online will be calculated. 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 = 400 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 = 400? Explain.