Using R for Introductory Statistics : John Verzani

Using R for introductory statistics 168* In Windows you may need to call flush, console() () to get the output. See the FAQ for details.6.5 Investigating distributions■ Example 6.1: The sample medianThe sample median, M, is a measurement of central tendency like the sample mean. Doesit, too, have an approximately normal distribution? How does the sampling distribution ofM reflect the parent distribution of the sample? Will M converge to some parameter ofthe parent distribution as converges to µ?Figure 6.4 Density estimates forsimulations of the sample medianfrom exponential data. As nincreases, the sampling distributionappears to become normallydistributed and concentrates on themedian of the parent population.To investigate these questions, we will perform a simulation. Assume we have a randomsample X 1 , X 2 , …, X n taken from the Exponential(1) distribution. This distribution hasmean 1 and median log(2) = .6931. We perform a simulation for n = 25, 100, and 400.First we define a function to find the median from the sample:> f = function(n) median(rexp(n))Now we generate samples for different sizes of n.> m = 500> res.25 = c(); res.100 = c(); res.400 = c()> ford in 1:m) res.25[i] = f(25)> for(i in 1:m) res.100[i] = f(100)> for(i in 1:m) res.400[i] = f(400)