EXAMPLE 5.5

Sample means are approximately Normal. Figure 5.6 illustrates two striking facts about the sampling distribution of a sample mean. Figure 5.6(a) displays the distribution of student visit lengths (in minutes) to a statistics help room at a large midwestern university. Students visiting the help room were asked to sign in upon arrival and then sign out when leaving. During the school year, there were 1838 visits to the help room but only 1264 recorded visit lengths. This is because many visiting students forgot to sign out. We also omitted a few large outliers (visits lasting more than 10 hours).5 The distribution is strongly skewed to the right. The population mean is μ = 61.28 minutes.

image
Figure 5.6 (a) The distribution of visit lengths to a statistics help room during the school year, Example 5.5. (b) The distribution of the sample means for 500 random samples of size 60 from this population. The scales and histogram classes are exactly the same in both panels.

295

Table : TABLE 5.1 Length (in Minutes) of 60 Visits to a Statistics Help Room
10 14 15 16 18 20 20 20 23 25
28 30 30 30 30 30 31 33 35 35
46 48 50 50 50 50 51 54 55 55
60 60 60 60 60 60 60 65 65 65
75 77 80 80 84 85 88 98 100 100
105 105 105 115 120 135 135 136 157 210

Table 5.1 contains the lengths of a random sample of 60 visits from this population. The mean of these 60 visits is = 63.45 minutes. If we were to take another sample of size 60, we would likely get a different value of . This is because this new sample would contain a different set of visits. To find the sampling distribution of , we take many SRSs of size 60 and calculate for each sample. Figure 5.6(b) is the distribution of the values of for 500 random samples. The scales and choice of classes are exactly the same as in Figure 5.6(a) so that we can make a direct comparison.

The sample means are much less spread out than the individual visit lengths. What is more, the Normal quantile plot in Figure 5.7 confirms that the distribution in Figure 5.6(b) is close to Normal.

image
Figure 5.7 Normal quantile plot of the 500 sample means in Figure 5.6(b). The distribution is close to Normal.