problem 3.23
a) add b to z values, what happens to theta
x<-c(1.82, 0.50,1.62,2.48,1.68,1.88,1.55,3.06,1.30)
y <-c(0.878,0.647,0.598,2.05,1.06,1.29,1.06,3.14,1.29)
z <-y-x
wilcox.test(z,alternative="two.sided",paired=F,conf.int=T,conf.level=0.95)

z <-y-x +5
wilcox.test(z,alternative="two.sided",paired=F,conf.int=T,conf.level=0.95)

new estimate is theta+b (where b is constant)

b) what if multiply by d, each Z
z <-y-x
wilcox.test(z,alternative="two.sided",paired=F,conf.int=T,conf.level=0.95)

z <-5*z
wilcox.test(z,alternative="two.sided",paired=F,conf.int=T,conf.level=0.95)

new estimate is theta+b (where b is constant)

c) for k, some positive, such that n>2k
z <-y-x
wilcox.test(z,alternative="two.sided",paired=F,conf.int=T,conf.level=0.95)
z_sort = sort(z)
let k=2  - then 9>2(2)

z_sort2 = z_sort[3:7]
wilcox.test(z_sort2,alternative="two.sided",paired=F,conf.int=T,conf.level=0.95)

for larger n, let k=25, then 100>2(25)
x1 <-rnorm(100,0,1)
y1 <-rnorm(100,2,1)
z1 <-y1-x1
wilcox.test(z1,alternative="two.sided",paired=F,conf.int=T,conf.level=0.95)
z1_sort= sort(z1)
k<-2
z1_sort2 = z1_sort[k+1:length(z1)-k+1]
wilcox.test(z1_sort2,alternative="two.sided",paired=F,conf.int=T,conf.level=0.95)



3.27
x=c(350,200,240,290,90,370,240)
y=c(480,130,250,310,280,1450,280)
z=y-x
z1=sort(z)
W=matrix(NA,length(z1),length(z1))
for (i in 1:length(z1))
{	for (j in 1:length(z1))
	{	if (i<=j)
		{	W[i,j]<-(z1[i]+z1[j])/2
		}
	}
}
W_sort= sort(W)
W_sort[3]
W_sort[26]

wilcox.test(z,alternative="two.sided",paired=F,conf.int=T,conf.level=0.954)