4.41. You can use an applet on a computer or on the Internet to repeatedly generate random samples from artificial populations and analyze them to study the properties of statistical methods. To try this, go to www.prenhall.com/agresti and use the sampling distribution applet. Select binary for the parent population, setting the population proportion as 0.50. Select for the sample size n = 100.

(a) Simulate once (setting the number of simulations N = 1 and clicking on Sample) and report the counts and the proportions for the two categories. Did you get a sample proportion close to 0.50? Perform this simulation of a random sample of size 100 ten times, each time observing from the graphs the counts and the corresponding sample proportion of yes votes. Summarize.

(b) Now plot the results of simulating a random sample of size 100 and finding the sample 4.42. Refer to the previous exercise.

(a) For this applet, select the skewed population distribution. Take 1000 samples of size 30 each. How does the empirical distribution of sample means compare to the population distribution? What does this reflect?

(b) Repeat, this time choosing a sample size of only 2 for each sample. Why is the sampling distribution not symmetric and bell-shaped?