The empirical rule states that the 1, 2, and 3 standard deviation intervals about the mean will contain 68%, 95%, and 99.7%, respectively.
a. Use the computer or calculator commands on page 90 to randomly generate a sample of 100 data from a normal distribution with mean 50 and standard deviation 10. Construct a histogram using class boundaries that are multiples of the standard deviation 10; that is, use boundaries from 10 to 90 in intervals of 10 (see the commands on pp. 52–54). Calculate the mean and the standard deviation using the commands found on pages 64 and 78; then inspect the histogram to determine the percentage of the data that fell within each of the 1, 2, and 3 standard deviation intervals. How closely do the three percentages compare to the percentages claimed in the empirical rule?
b. Repeat part a. Did you get results similar to those in part a? Explain.
c. Consider repeating part a several more times. Are the results similar each time? If so, in what way?
d. What do you conclude about the truth of the empirical rule?