C.10 This exercise is a low-tech simulation experiment related to Exercise C.9. It can be a grcup or class exercise if desired. Have each group member create a set of 10 numbered, identical, slips of paper like the following table.

(a) Draw a slip of paper at random and record its value, preferably entering each number into a data file for use with your computer software. Draw a total of 10 times, each time replacing the slip into the pile and stirring them well. Compare the average of these values to the expected value in Exercise C.9(b). Draw 10 more values with replacement. What is the average of all 20 values?

(b) Calculate the sample variance of the 20 values obtained in part (a). Compare this value to the true variance in Exercise C.9(c).

(c) Draw three slips of paper at random, with replacement. Calculate the average of the numbers on these N : 3 slips of paper, V : (\ t yz -t I3)/3. Repeat this process at least NSAM:20times, obtaining NSAM average values, V ,,V z, . . . ,Y *r*. Calculate the sample average and sample variance of these NSAM values. Compare these to the expected value and variance of the sample mean obtained in Exercise C.9(d).

(d) Enter the NSAM values7 1 ,72,...,Vrro, into a data file. Standardizet hese values by subtracting the true mean and dividing by the true standard deviation of the mean, from Exercise C9(d). Use your computer software to create a histogram. Discuss the central limit theorem and how it relates to the figure you have created.

(e) Repeat parts

(c) and

(d) using NSAM samples of more than N: 3 slips of paper, perhaps 5 or 7. How do the histograms compare to the one in parf (d)?

(f) Discuss the terms "sampling variation" and "sampling distribution" in the context of the experiments you have performed.

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