This exercise is designed to give you a more direct understanding of the standard error of the mean. Create a population as follows: Get 20 identical small slips of paper or file cards. On eight of these slips, write the number 50; on five of the slips, write the number 51; on another five, write the number 49. For the last two slips of paper, write the number 48 on one and 52 on the other. Place all the slips in a bowl and mix thoroughly.

a. Draw one slip at random from the bowl and write down its number. Then replace the slip in the bowl, mix thoroughly, and draw at random again (this is called sampling with replacement). Keep repeating this process until you have written down the numbers for five random selections. Calculate the average for these five numbers. This is your first sample mean.

b. Repeat the process described in part (a) until you have calculated a total of six sample means.

c. Calculate the mean and (unbiased) SD of the six sample means you found in part (b). The latter statistic is the standard error of the mean calculated directly from the sample means rather than estimated from a sample SD divided by N. Is the mean of the sample means about what you expected?

d. Estimate the standard error of the mean separately from the SD of each of the six samples you drew. How do these six estimates compare to each other, and to the standard error you calculated directly in part (c)?