This exercise will give you a tangible understanding of sampling distributions and the two theorems presented in this chapter. For this demonstration only, the 2012 GSS database will be treated as a population and its characteristics will be treated as parameters.

We will use SPSS to draw ten random samples and calculate statistics for age on each sample. Recall that a sampling distribution includes all possible samples, so these ten samples are the merest beginning of an actual sampling distribution.

After drawing each sample, you will record the mean and the standard error of the mean for age. Recall that the standard error is the standard deviation of the sampling distribution (sX). The means of our ten samples should be within a standard error of the population mean, which in this artificial situation we know is 48.21

(m 5 48.21).

● Find and click the SPSS icon on your desktop.

● Load the GSS2012 data set.

● Click Data from the menu bar of the Data Editor and then click Select Cases. Next, click the button next to Random sample of cases and then click the Sample button. On the “Select Cases: Random Sample” window, we can specify the size of the sample in two different ways. We’ll use the second option and select samples of size 200 (N = 200).

Click the button next to Exactly and fill in the boxes so that this line reads “Exactly 200 cases from the first 1457 cases.”

● Click Continue, and then click OK on the “Select Cases” window. The sample will be selected.

● To find the mean age for the sample, click Analyze, Descriptive Statistics, and Descriptives. Find age in the variable list and transfer it to the

“Variable(s)”: window.

● On the “Descriptives” menu, click the Options button and select S.E. MEAN in addition to the usual statistics. Click Continue and then OK, and the requested statistics will appear in the output window.

● Write your results in the following table. I have entered the results from our first sample.

● Before getting the next sample, we need to restore the full data set (N = 1457). To do this, click Data, Select Cases, and then click the Reset button at the bottom of the window. Click OK and the GSS data set will be restored to its original size.

● To produce the second sample, click Data, Select Cases, and then click the button next to Random Sample of Cases. Click Sample and, as before, specify “Exactly 200 cases from the first 1457 cases.”

● Click Continue and OK, and the second sample will be selected. Use Descriptives to calculate statistics and record your results in the following table.
● Repeat these steps for the third sample and record your results. Make sure that you restore the full data set (click Data, Select Cases, and Reset)
each time before selecting the next sample.
Continue until you have results for ten samples.
To complete the table ● Enter the sample means and standard errors (S.E.)
of the mean in the appropriate columns for each sample.
● The right-hand column shows an interval that is the sample mean plus and minus the standard error. Find the lower limit of the interval by subtracting the standard error from the sample mean and the upper limit by adding the standard error to the sample mean. The population mean (m = 48.21) will almost always be in this interval.
X S.E. X 6 S.E.
1.47.81 1.2946.52 to 49.10 2. _____ to _____ 3. _____ to _____ 4. _____ to _____ 5. _____ to _____ 6. _____ to _____ 7. _____ to _____ 8. _____ to _____ 9. _____ to _____ 10. _____ to _____ What can we conclude? First, if you had the patience to draw all ten samples, congratulations! Second, I must point out that this exercise would work better if you drew even more samples. Remember that the theorems about the characteristics of the sampling distribution are based on an infinite number of sample outcomes.

Third, the ten outcomes in the table are a sampling distribution, although not a very developed one. The ten outcomes in the table are just the beginning of a very long process if we were to construct an actual sampling distribution.

Remember that the sampling distribution is normal and has a mean equal to the population mean and a standard deviation equal to the standard error (S.E.). About 68%

of all sample outcomes will be within 61 standard error of the mean, 95% will be within 62 standard errors, and so forth. What percentage of your ten sample means are in these ranges? Random chance is at work here and some of your sample means might be very different from the population mean, but most should be reasonably close (i.e., within 61 S.E.).

We would not ordinarily know the value of the population mean, but we do know that it is the same value as the mean of the sampling distribution. Thus, the odds are in our favor that any particular sample mean will be close to the parameter. This exercise should convince you that properly drawn EPSEM samples are very likely to be representative of populations, and statistics computed on these samples should be close in value to their respective parameters.