Using R for Introductory Statistics : John Verzani

Simulation 1756.3 For what value of n does look approximately normal when each is Uniform(0,1)? (Run several simulations for different n’s and decide where the switch to normalitybegins.)6.4 For what value of n does look approximately normal when each Xi is(Exponential(1) (rexp(n, 1))?6.5 For what value of n does look approximately normal when each Xi has a t-distribution with 3 degrees of freedom (rt (n, 3))?6.6 Compare the distributions of the sample mean and sample median when the X ihave the t distribution with 3 degrees of freedom and n=10. Which has a bigger spread?6.7 The χ 2 distribution arises when we add a number of independent, squared standardnormals. Instead of using rchisq() to generate samples, we can simulate by addingnormally distributed numbers. For example, we can simulate a χ 2 distribution with 4degrees of freedom with> res = c()> for(i in 1:500) res[i] = sum(rnorm(4)^2)> qqnorm(res)Repeat the above for 10, 25, and 50 degrees of freedom. Does the data ever appearapproximately normal? Why would you expect that?6.8 The correlation between and s 2 depends on the parent distribution. For a normalparent distribution the two are actually independent. For other distributions, this isn’t so.To investigate, we can simulate both statistics from a sample of size 10 andobserve their correlation with a scatterplot and the cor() function.> xbar = c();std = c()> for(i in 1:500) {+ sam = rnorm(10)+ xbar [i] = mean(sam); std[i] = sd(sam)+}> plot(xbar,std)> cor(xbar,std)[1] 0.09986The scatterplot (not shown) and small correlation is consistent with known independenceof the variables.Repeat the above with the t-distribution with 3 degrees of freedom (a longtailedsymmetric distribution) and the exponential distribution with rate 1 (a skeweddistribution). Are there differences? Explain.6.9 For a normal population the statistic has a normaldistribution. LetThat is, σ is replaced by s, the sample standard deviation. The sampling distribution of Tis different from that of Z.