For n observations {yi}, let y(1) ≤ y(2) ≤ ⋯ ≤ y(n) denote theirorderedvalues,called order statistics. Let qi bethe i~(n + 1) quantileofthestandardnormaldistribution,for i = 1. ...,n. When {yi} are arandomsamplefromanormaldistribution,theplotofthepoints

(q1, y(1)), ..., (qn, y(n)) should approximatelyfollowastraightline,morecloselysowhen n is large. This normal quantileplot is aspecialcaseofa quantile-quantile (Q-Q)plot. The R appendixofthisbookpresentsdetails.

(a) Randomlygenerate(i) n = 10, (ii) n = 100, (iii) n = 1000 observationsfroma N(0, 1)

distribution andconstructthenormalquantileploteachtime,usingsoftwaresuchasthe R functions rnorm and qqnorm. Notethatas n increases thepointsclustermoretightly along theline y = x, whichyoucanaddtotheplotwithcommand abline(0, 1).

(b) Randomlygenerate1000observationsfroma N(100, 162) distribution ofIQ’sandcon-

struct thenormalquantileplot.Whatistheslopeofthelineapproximatingthesepoints?

(c) Randomlygenerate1000observationsfromthe(i)exponentialdistribution(2.2),(ii)uni-

form distributionover(0,1),usingsoftwaresuchasthe R functions rexp and runif. Con-

struct thenormalquantileplotineachcase.Explainhowtheyrevealthenon-normality of thedata.

(d) Forcase(ii)in(c),findappropriateuniformquantilesforwhichthe Q-Q plot wouldbe approximatelylinear.Constructtheplot.