152 PART II INFERENTIAL STATISTICS sampling be an appropriate technique (assuming that no list was available)? Describe in detail how means should be very close in value because Ux = M by the Central Limit Theorem. you would construct a cluster sample. c. Calculate the standard deviation of the sam- pling distribution (use the means as scores) 6.2 This exercise is extremely tedious and hardly ever and the standard deviation of the population. works out the way it ought to, mostly because not Compare these two values. You should find many people have the patience to draw an "infi- that ox = o/ VN. nite" number of even very small samples. How- d. If none of the preceding exercises turned out ever, if you want a more concrete and tangible un- as they should have, it is for one or more of the derstanding of sampling distributions and the two following reasons: theorems presented in this chapter, then this exer- 1. You didn't take enough samples. You may cise may have a significant payoff. At the end of need as many as 100 or 200 (or more) this problem are listed the ages of a population of samples to see the curve begin to look college students (N = 50). By a random method "normal." (such as a table of random numbers), draw at least 2. Sample size (2) is too small. An N of 5 or 50 samples of size 2 (that is, 50 pairs of cases), com- 10 would work much better. pute a mean for each sample, and plot the means on a frequency polygon. (Incidentally, this exer- 3. Your sampling method is not truly random cise will work better if you draw 100 or 200 and/or the population is not arranged in samples and/or use larger samples than N = 2.) random fashion. a. The curve you've just produced is a sampling 17 20 20 19 20 distribution. Observe its shape; after 50 18 21 19 20 19 samples, it should be approaching normality. 19 22 19 23 19 What is your estimate of the population mean 20 23 18 20 20 (u) based on the shape of the curve? 22 19 19 20 20 b. Calculate the mean of the sampling distribu- 23 17 18 21 20 tion (My). Be careful to do this by summing the 20 18 20 19 20 sample means (not the scores) and dividing by 17 21 21 21 20 20 22 the number of samples you've drawn. Now 18 21 20 2 21 compute the population mean (M). These two SPSS for Windows Using SPSS for Windows to Draw Random Samples Average Age