Question

4.2.12. In Exercise 4.2.11, the sampling was from the N(0,1) distribution. Show, however, that setting μ = 0 andσ = 1 is without loss of generality. Hint: First,X1,...,Xn is a random sample from the N(μ,σ2) if and only if Z1,...,Zn is a random sample from the N(0,1), where Zi =( Xi − μ)/σ. Then show the confidence interval based on the Zi’s contains 0 if and only if the confidence interval based on the Xi’s contains μ.

For more info about 4.2.11 :

(4.2.11. Let X1,...,Xn be a random sample from a N(0,1) distribution. Then the probability that the random interval X±tα/2,n−1(s/√n) trapsμ = 0 is (1−α). To verify this empirically, in this exercise, we simulate m such intervals and calculate the proportion that trap 0, which should be “close”to (1−α). (a) Set n = 10 andm = 50. Run the R code mat=matrix(rnorm(m*n),ncol=n) which generates m samples of size n from the N(0,1) distribution. Each row of the matrix mat contains a sample. For this matrix of samples, the function below computes the (1 − α)100% confidence intervals, returning them in a m×2 matrix. Run this function on your generated matrix mat. What is the proportion of successful confidence intervals?
246 Some Elementary Statistical Inferences
getcis <- function(mat,cc=.90){ numb <- length(mat[,1]); ci <- c() for(j in 1:numb) {ci<-rbind(ci,t.test(mat[j,],conf.level=cc)$conf.int)} return(ci)} This function is also at the site discussed in Section 1.1.
(b) Run the following code which plots the intervals. Label the successful intervals. Comment on the variability of the lengths of the confidence intervals. cis<-getcis(mat); x<-1:m plot(c(cis[,1],cis[,2])~c(x,x),pch="",xlab="Sample",ylab="CI") points(cis[,1]~x,pch="L");points(cis[,2]~x,pch="U"); abline(h=0))

Answer 1

n=10 m=50 matematrix(rnorm(m®n), ncol=n) getcis = function(mat, CC=0.90){ numb = length(mat[,2]) ci = c() for(j in 1:numb){ ci = rbind(ci, t.test(mat[j,), conf.level = cc)$conf.int) return(ci) getcis(mat) # # [,1] [,2] ## [1,] -0.727142060 0.77791436 ## [2,] -0.563145924 0.73195711 ## [3,] -0.462069623 0.97057237 ## [4,] -0.583300719 0.40439492 ## [5,] -0.907776166 0.25557473 [6,] -0.464229553 0.79127665 [7,1 -0.569700427 0.11455682 [8,] -0.506673394 0.87057980 [9,] -0.840218696 0.13187937 ## [10,] -0.513992272 0.43147450 ## 11,1 -0.592976912 0.34132995 ## [12,1 0.459364183 1.18616941 (13,] -1.163672672 -0.14977873 ## [14, ] -0.637146153 0.59455596 ## [15, ] -0.377295443 0.57680825 ## [16,] -0.765893195 0.12198084 ## [17,] -0.643764751 0.44201980 ## (18,] -0.506104936 0.74373663 ## [19,] -0.102022215 1.19812803 ## [20,1 -0.335999692 0.77818433

# # # # # # # # # # # # # # # # # # ########### ## [21] -1.288379458 8.31164588 ## [22, 1 3 ,869876471 1.54295117 ## [23] -8.541871491 8.20973329 ## [24] -0.764176274 8.78843688 ## [25] -8.49398281 1.39816455 ##「26.1 -0.634987978 8.91933817 ## 27, -8.519263739 8.52912392 [28] -.572211863 6.53878682 [29] -.415658772 8.82145333 ##[3,] -0.577839395 .66232399 ## [31] -8.495844143 8.86876396 ## [32] 8.062797631 8.85979444 ##[3,] -0.661164136 -0.89818864 #[34] -.462549466 1.88567232 ## [35] -8.271975354 8.96427493 [36] -.347841666 8.85324351 [37] -8.881695833 8.51782761 [38] -8.119873221 1.18622541 [39] -0.657777784 8.47468153 [48] -0.891848865 6,47982766 [41] -1.887887763 6.66595866 ## [42] -8.697952468 8.55359948 ## [43] -8.818828488 8.29859951 ## [44] -.398126114 1.17631661 ## [45] -1.855837814 -0.89659735 [46] -8.475889749 8.54684895 [47] -8.521652118 8.43791782 [48] -0.485163938 .27966675 [49] -.998983687 8.25279414 ##[50,] -0.878487841 8.83758996 # # # # # # #

cis<-getcis(mat) X<-1:m plot(c(cis[,1],cis[,2])-C(x,x), pch="",xlab="Sample", ylab="CI") points(cis[,1]-X, pch="L"); points(cis[,2]-X,pch="U") abline(h=0) U U 1.0 uu uu yw you... CY w 0.0 UUU U LU -1.0 0 10 20 30 40 50 Sample