Refer to the previous exercise. Using this applet, let’s check that the confidence interval for a proportion may work poorly with small samples. Set the population proportion π = 0.10, with n = 10. Draw 100 random samples, each of size 10, forming 95% confidence intervals for

π for each one.

(a) How many intervals failed to contain the true value,

π = 0.10? How many would you expect not to contain the true value?What does this suggest? (Notice that many of the intervals contain only the value 0.0, which happens when πˆ = 0.0.)

(b) To see that this is not a fluke, reset and draw 1000 confidence intervals. What percentage contain π = 0.10?

(Note: For every interval formed, the number of “successes”

is smaller than 15, so the large-sample formula is not adequate.)

(c) Using the Sampling Distribution of the Sample Proportion applet at www.artofstat.com/webapps.html, select a population proportion of 0.10.

Draw 10,000 random samples of size 10 each. Look at the empirical sampling distribution of the sample proportion values. Is it bell shaped and symmetric? Use this to help explain why the confidence interval performs poorly in this case.