The goal of this mini-project is to help you verify the Rule for Sample Proportions firsthand, using a physical simulation. You will use the population represented in Figure 19.1 to do so. It contains 400 individuals, of whom 160 (40%) are—that is, carry the gene for a disease—and the remaining 240 (60%) are—that is, do not carry the gene. You are going to draw 20 samples of size 15 from this population. Here are the steps you should follow:
Step 1: Develop a method for drawing simple random samples from this population. One way to do this is to cut up the symbols and put them all into a paper bag, shake well, and draw from the bag. There are less tedious methods, but make sure you actually get random samples. Explain your method.
Step 2: Draw a random sample of size 15 and record the number and percentage who carry the gene.
Step 3: Repeat step 2 a total of 20 times, thus accumulating 20 samples, each of size 15. Make sure to start over each time; for example, if you used the method of drawing symbols from a paper bag, then put the symbols back into the bag after each sample of size 15 is drawn so they are available for the next sample as well.
Step 4: Create a stemplot or histogram of your 20 sample proportions. Compute the mean.
Step 5: Explain what the Rule for Sample Proportions tells you to expect for this situation.
Step 6: Compare your results with what the Rule for Sample Proportions tells you to expect. Be sure to mention mean, standard deviation, shape, and the intervals into which you expect 68%, 95%, and almost all of the sample proportions to fall.