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.

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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.