#4.1

y<-c(1651,1112, 102.4,100,67.7,65.9,64.7,39.6,31)
x<- c(48.1,48,45.5,41.7,35.4,34.3,32.4,29.1,27.3,18.9,6.6,5.2,4.7)
n =min(length(x),length(y))
N = length(x) + length(y)
ans = wilcox.test(y,x,alternative="less",paired=F,exact=T)
W = ans$statistic + n*(n+1)/2
W

w = c(x,y)
W_1 = 0
w_sort = sort(w)
x1 = rank(w_sort)
for (i in 1:N)
{	for (j in 1:length(y))
	{	if (w_sort[i] == y[j])
		{	W_1 = x1[i] + W_1
			break
		}	
	}
}
W_1 


#4.5
y<-c(37,39,30,7,13,129,45,25,16,146,94,16,23,1,290,169,62,145,36,20,13)
x<-c(12,44,34,14,9,19,156,23,13,11,47,26,14,33,15,62,5,8,0,154,146)
n =min(length(x),length(y))
N = length(x) + length(y)
ans = wilcox.test(y,x,alternative="greater",paired=F,exact=T)
W = ans$statistic + n*(n+1)/2
W
EW = n*(N+1)/2
VarW = length(x)*length(y)*(N+1)/12
Z = (W - EW)/(VarW)^0.5
1-pnorm(Z)


#not working, b/c of ties
w = c(x,y)
W_1 = 0
w_sort = sort(w)
x1 = rank(w_sort)
for (i in 1:N)
{	for (j in 1:length(y))
	{	if (w_sort[i] == y[j])
		{	W_1 = x1[i] + W_1
			break
		}	
	}
}
W_1 



#4.15
y<-c(1651,1112, 102.4,100,67.7,65.9,64.7,39.6,31)
x<- c(48.1,48,45.5,41.7,35.4,34.3,32.4,29.1,27.3,18.9,6.6,5.2,4.7)
length(y)
x1<-y[1]-x
x2<-y[2]-x
x3<-y[3]-x
x4<-y[4]-x
x5<-y[5]-x
x6<-y[6]-x
x7<-y[7]-x
x8<-y[8]-x
x9<-y[9]-x
y1=c(x1,x2,x3,x4,x5,x6,x7,x8,x9)
y1_sort = sort(y1)
median(y1_sort)
mean(y)-mean(x)



#4.23
y<-c(1651,1112, 102.4,100,67.7,65.9,64.7,39.6,31)
y1<-c(1651,1112, 1024,100,67.7,65.9,64.7,39.6,31)
x<- c(48.1,48,45.5,41.7,35.4,34.3,32.4,29.1,27.3,18.9,6.6,5.2,4.7)
length(y)
x1<-y[1]-x
x2<-y[2]-x
x3<-y[3]-x
x4<-y[4]-x
x5<-y[5]-x
x6<-y[6]-x
x7<-y[7]-x
x8<-y[8]-x
x9<-y[9]-x
ynew1=c(x1,x2,x3,x4,x5,x6,x7,x8,x9)
ynew1_sort = sort(ynew1)
theta1 = median(ynew1_sort)
mean1 = mean(y)-mean(x)

x1<-y1[1]-x
x2<-y1[2]-x
x3<-y1[3]-x
x4<-y1[4]-x
x5<-y1[5]-x
x6<-y1[6]-x
x7<-y1[7]-x
x8<-y1[8]-x
x9<-y1[9]-x
ynew2=c(x1,x2,x3,x4,x5,x6,x7,x8,x9)
ynew2_sort = sort(ynew2)
theta2 = median(ynew2_sort)
mean2 = mean(y1)-mean(x)

theta1
theta2
mean1
mean2


$4.24
#part a
y<-c(1651,1112, 102.4,100,67.7,65.9,64.7,39.6,31)+4
x<- c(48.1,48,45.5,41.7,35.4,34.3,32.4,29.1,27.3,18.9,6.6,5.2,4.7)+6
length(y)
x1<-y[1]-x
x2<-y[2]-x
x3<-y[3]-x
x4<-y[4]-x
x5<-y[5]-x
x6<-y[6]-x
x7<-y[7]-x
x8<-y[8]-x
x9<-y[9]-x
y1=c(x1,x2,x3,x4,x5,x6,x7,x8,x9)
y1_sort = sort(y1)
median(y1_sort)
#theta = thetaold + 4-6


#part b
y<-c(1651,1112, 102.4,100,67.7,65.9,64.7,39.6,31)
y = 2*y
x<- c(48.1,48,45.5,41.7,35.4,34.3,32.4,29.1,27.3,18.9,6.6,5.2,4.7)
x = 2*x
x1<-y[1]-x
x2<-y[2]-x
x3<-y[3]-x
x4<-y[4]-x
x5<-y[5]-x
x6<-y[6]-x
x7<-y[7]-x
x8<-y[8]-x
x9<-y[9]-x
y1=c(x1,x2,x3,x4,x5,x6,x7,x8,x9)
y1_sort = sort(y1)
median(y1_sort)
#theta = 2*thetaold


#4.27
y<-c(1651,1112, 102.4,100,67.7,65.9,64.7,39.6,31)
x<- c(48.1,48,45.5,41.7,35.4,34.3,32.4,29.1,27.3,18.9,6.6,5.2,4.7)
x1<-y[1]-x
x2<-y[2]-x
x3<-y[3]-x
x4<-y[4]-x
x5<-y[5]-x
x6<-y[6]-x
x7<-y[7]-x
x8<-y[8]-x
x9<-y[9]-x
y1=c(x1,x2,x3,x4,x5,x6,x7,x8,x9)
y1_sort = sort(y1)
median(y1_sort)
LCL = y1_sort[29]
LCL
UCL = y1_sort[89]
UCL