Results for Additional Exercise 2 for Session 2.

pnorm(-qnorm(.975)+sqrt(15/2)*1/1)
pt(qt(.975,28),df=28,ncp=sqrt(15/2)*1/1,lower.tail=FALSE)

library(MASS)

# reproduces the values in the table on p. 29 (compound symmetry (rho))
x <- matrix(c(1,1,1,0,2,5),nrow=2,ncol=3,byrow=TRUE)
n.list <- c()
for (i in 0:2) {
  r <- 0.2+0.3*i
  e <-
  ginv(x%*%ginv(matrix(c(1,r,r,r,1,r,r,r,1),nrow=3,ncol=3))%*%t(x))[2,2]
  for (j in 1:3) {
    s2 <- j*100
    n.list <- c(n.list,2*(qnorm(.05)+qnorm(.2))^2*e*s2/.5^2)
    }
  }
n.list
[1] 312.38184 624.76368 937.14552 195.23865 390.47730 585.71595  78.09546 156.19092 234.28638

# table for 1st order autoregressive (rho)
n.list <- c()
for (i in 0:2) {
  r <- 0.2+0.3*i
  e <-
  ginv(x%*%ginv(matrix(c(1,r,r^2,r,1,r,r^2,r,1),nrow=3,ncol=3))%*%t(x))[2,2] 
  for (j in 1:3) {
    s2 <- j*100
    n.list <- c(n.list,2*(qnorm(.05)+qnorm(.2))^2*e*s2/.5^2)
    }
  }
n.list
[1]  373.4542  746.9085 1120.3627  289.8074  579.6147  869.4221  137.9320 275.8640  413.7959

# reproduces the values in the table on p. 30 (compound symmetry (rho))
x <- rep(c(1),3)
n.list <- c()
for (i in 0:2) {
  r <- 0.2+0.3*i
  e <- t(x)%*%ginv(matrix(c(1,r,r,r,1,r,r,r,1),nrow=3,ncol=3))%*%x
  for (j in 0:3) {
    d <- (20+j*10)/100
    n.list <- c(n.list,2*(qnorm(.05)+qnorm(.2))^2/d^2/e)
    }
  }
n.list
[1] 144.25967  64.11541  36.06492  23.08155 206.08524  91.59344  51.52131 32.97364 267.91081 119.07147  66.97770  42.86573

# table for 1st order autoregressive (rho)
x <- rep(c(1),3)
n.list <- c()
for (i in 0:2) {
  r <- 0.2+0.3*i
  e <- t(x)%*%ginv(matrix(c(1,r,r^2,r,1,r,r^2,r,1),nrow=3,ncol=3))%*%x 
  for (j in 0:3) {
    d <- (20+j*10)/100
    n.list <- c(n.list,2*(qnorm(.05)+qnorm(.2))^2/d^2/e)
    }
  }
n.list
[1] 132.48337  58.88150  33.12084  21.19734 185.47672  82.43410  46.36918 29.67627 252.92280 112.41013  63.23070  40.46765