Using R for Introductory Statistics : John Verzani

Describing populations 157Figure 5.8 Density of for n=5 andn=25 along with parent populationNormal(0, 1). As n increases, thedensity concentrates on µ.chosen males is again normal with mean 70.2 but standard deviation 1/5 as large. Theprobability that the sample average is between 70 and 71 is found with> mu=70.2; sigma=2.89; n=25> diff( pnorm(70:71, mu, sigma/sqrt(n)) )[1] 0.5522Compare this to the probability for a single person> diff( pnorm(70:71, mu, sigma) )[1] 0.13665.3.2 Nonnormal parent populationThe central limit theorem states that for any parent population with mean µ and standarddeviation σ, the sampling distribution of for large n satisfieswhere Z is a standard normal random variable. That is, for n big enough, the distributionof once standardized is approximately a standard normal distribution. We also refer tothis as saying is asymptotically normal.Figure 5.9 illustrates the central limit theorem for data with an Exponential (1)distribution. This parent population and simulations of the distribution of for n=5, 25,and 100 are drawn. As n gets bigger, the sampling distribution of becomes more andmore bell shaped.