EXAMPLE 5.24

Consider a simulation experiment in which the population distribution is quite skewed. Figure 5.13 shows the density curve for lifetimes of a certain type of electronic control

• this is actually a lognormal distribution with $E\left(\ln \left(X\right)\right)=3$ and $V\left(\ln \left(X\right)\right)=.16$

Again the statistic of interest is the sample mean $\bar{X}$. The experiment utilized 500 replications and considered the same four sample sizes as in Example 5.23. The resulting histograms along with a normal probability plot from Minitab for the $500\bar{x}$ values based on $n=30$ are shown in Figure 5.14.

Figure 5.13 Density curve for the simulation experiment of Example 5.24 $E\left(X\right)=21.7584$ , $V\left(X\right)=82.1449$

Figure 5.14 Results of the simulation experiment of Example 5.24: (a) $\bar{x}$ histogram for $n=5$ ; (b) $\bar{x}$ histogram for $n=10$ ; (c) $\bar{x}$ histogram for $n=20$ ; (d) $\bar{x}$ histogram for $n=30$ ; (e) normal probability plot for $n=30$ (from Minitab)

Unlike the normal case, these histograms all differ in shape. In particular, they become progressively less skewed as the sample size $n$ increases. The average of the $500\bar{x}$ values for the four different sample sizes are all quite close to the mean value of the population distribution. If each histogram had been based on an unending sequence of $\bar{x}$ values rather than just 500, all four would have been centered at exactly 21.7584. Thus different values of $n$ change the shape but not the center of the sampling distribution of $\bar{X}$. Comparison of the four histograms in Figure 5.14 also shows that as $n$ increases, the spread of the histograms decreases. Increasing $n$ results in a greater degree of concentration about the population mean value and makes the histogram look more like a normal curve. The histogram of Figure 5.14(d) and the normal probability plot in Figure 5.14(e) provide convincing evidence that a sample size of $n=30$ is sufficient to overcome the skewness of the population distribution and give an approximately normal $\bar{X}$ sampling distribution.