8.28 Simulating confidence intervals with poor coverage Using the Explore Coverage web app, let’s check that the large-sample confidence interval for a proportion may work poorly with small samples. In the app, set p = 0.30, n = 10 and leave the confidence level at 95%. Select to draw 100 random samples of size n and then click on Draw Sample(s).

a. How many of the intervals you generated with the app fail to contain the true value, p = 0.30?

b. How many would you expect not to contain the true value? What does this suggest?

c. To see that this is not a fluke, now take 1000 samples and see what percentage of 95% confidence intervals contain 0.30. (Note: For every interval formed, the number of successes is smaller than 15, so the large-sample formula is not adequate.)

d. Using the Sampling Distribution for a Sample Proportion web app, generate 10,000 random samples of size 10 when p = 0.30. The app will plot the simulated sampling distribution of the sample proportion values. Is it bell shaped? Use this to help you explain why the large-sample confidence interval performs poorly in this case. (This exercise illustrates that assumptions for statistical methods are important, because the methods may perform poorly if we use them when the assumptions are violated.)