EXAMPLE 5.23

The population distribution for our first simulation study is normal with $\mu =8.25$ and $\sigma =.75$ , as pictured in Figure 5.11. [The article “Platelet Size in Myocardial Infarction” (British Med. J., 1983: 449-451) suggests this distribution for platelet volume in individuals with no history of serious heart problems.]

Figure 5.11 Normal distribution, with $\mu =8.25$ and $\sigma =.75$

We actually performed four different experiments, with 500 replications for each one. In the first experiment,500 samples of $n=5$ observations each were generated using Minitab, and the sample sizes for the other three were $n=10,n=20$ , and $n=30$ , respectively. The sample mean was calculated for each sample, and the resulting histograms of $\bar{x}$ values appear in Figure 5.12.

Figure 5.12 Sample histograms for $\bar{x}$ based on 500 samples, each consisting of $n$ observations: (a) $n=5$ ; (b) $n=10$ ; (c) $n=20$ ; (d) $n=30$

The first thing to notice about the histograms is their shape. To a reasonable approximation, each of the four looks like a normal curve. The resemblance would be even more striking if each histogram had been based on many more than $500\bar{x}$ values. Second, each histogram is centered approximately at 8.25 , the mean of the population being sampled. Had the histograms been based on an unending sequence of $\bar{x}$ values, their centers would have been exactly the population mean,8.25 .

The final aspect of the histograms to note is their spread relative to one another. The larger the value of $n$ , the more concentrated is the sampling distribution about the mean value. This is why the histograms for $n=20$ and $n=30$ are based on narrower class intervals than those for the two smaller sample sizes. For the larger sample sizes, most of the $\bar{x}$ values are quite close to 8.25 . This is the effect of averaging. When $n$ is small, a single unusual $x$ value can result in an $\bar{x}$ value far from the center. With a larger sample size, any unusual $x$ values, when averaged in with the other sample values, still tend to yield an $\bar{x}$ value close to $\mu$ . Combining these insights yields a result that should appeal to your intuition: $\bar{X}$ based on a large $n$ tends to be closer to $\mu$ than does $\bar{X}$ based on a small $n$ .