Using R for Introductory Statistics : John Verzani

Using R for introductory statistics 1646.3 Simulations related to the central limit theoremWe use a for loop to investigate the normality of for different parent populations anddifferent sample sizes. For example, if the X i are Uniform(0, 1) we can simulate forn=2, 10, 25, and 100 with these commands:## set up plot window> plot(0,0,type="n",xlim=c(0,1),ylim=c(0,13.5),+ xlab="Density estimate",ylab="f(x)")> m = 500;a=0;b=1> n = 2> for (i in 1:m) res[i]=mean(runif(n,a,b))> lines(density(res),lwd=2)## repeat last 3 lines with n=10, 25, and 100Figure 6.3 Density estimates for forn=2, 10, 25, and 100 with Uniform(0,1) dataIn Figure 6.3 a density estimate is plotted for each simulation. Observe how the densitiessqueeze in and become approximately bell shaped, as expected, even for n=10. As thestandard deviation of is if n goes up four times (from 25 to 100, for example),the standard deviation gets cut in half. Comparing the density estimate for n=25 andn=100, we can see that the n=100 graph has about half the spread.In this example the for loop takes the shortened formfor(i in values) a_single_commandIf there is just a single command, then no braces are necessary. This is convenient whenwe use the up arrow to edit previous command lines.In the problems, you are asked to simulate for a variety of parent populations toverify that the more skewed the data is, the larger n must be for the normal distribution toapproximate the sampling distribution of