#
# Written by:
# -- 
# John L. Weatherwax                2009-04-21
# 
# email: wax@alum.mit.edu
# 
# Please send comments and especially bug reports to the
# above email address.
# 
#-----

# Problem 1 (epage 152): 
# 
Xbar = 78
mu = 80
sigma = 5
n = 10
Z = ( Xbar - mu)/(sigma/sqrt(n))
alpha = 0.05
c = qnorm( alpha )

# Problem 2-3 (epage 152): 
# 
c = qnorm( 1 - 0.5*alpha )

Xbar - c * sigma/sqrt(n)
Xbar + c * sigma/sqrt(n)

# Problem 4-5 (epage 152): 
# 
c = ( Xbar - mu)/(sigma/sqrt(n))
p = pnorm( c )

p = 2 * pnorm( c )

# Problem 6-7 (epage 152):
#
xbar = 120
sigma = 5
n = 49
alpha = 0.05
mu = 130

Z = ( xbar - mu )/(sigma/sqrt(n))

c = qnorm(alpha)
if( Z < c ){
  print("reject H_0")
}else{
  print("cannot reject H_0")
}

c = -qnorm(0.5*alpha) # for a two-sided test

xbar - c * sigma/sqrt(n)
xbar + c * sigma/sqrt(n)

# Problem 10 (epage 153):
#
mu = 232
xbar = 240
sigma = 4
n = 25
alpha = 0.01

Z = ( xbar - mu )/(sigma/sqrt(n))

c = -qnorm( alpha )

if( Z <= c ){
  print("cannot reject H_0")
}else{
  print("must reject H_0")
}

# Problem 11 (epage 153):
#
xbar = 565
mu = 546
sigma = 40
n = 20
alpha = 0.05

Z = ( xbar - mu )/(sigma/sqrt(n))

c = qnorm( 1 - 0.5*alpha )

xbar - c * sigma/sqrt(n) # the confidence interval 
xbar + c * sigma/sqrt(n)


# Problem 13 (epage 153):
#
alpha = 0.01
c = qnorm( alpha )
n = 25
mu = 56
mu_0 = 60
sigma = 5 
pnorm( c - ( sqrt(n)*( mu - mu_0 ) )/sigma )

# Problem 14 (epage 153):
#
alpha = 0.025
c = qnorm( 1 - alpha )
n = 36
mu = 103
mu_0 = 100
sigma = 8 
1 - pnorm( c - ( sqrt(n)*( mu - mu_0 ) )/sigma )

# Problem 15 (epage 153):
# Not sure that this is correct???
# 
alpha = 0.05
c1 = qnorm( 0.5*alpha )
c2 = qnorm( 1 - 0.5*alpha )
n = 49
mu = 47
mu_0 = 50
sigma = 10
pnorm( c1 - sqrt(n)*abs( mu - mu_0 )/sigma )
pnorm( -c2 - sqrt(n)*abs( mu - mu_0 )/sigma )

# Problem 16,17,18 (epage 153):
# 
alpha = 0.05
xbar = 46
mu = 48 
n = 10
sigma = 5
Z = ( xbar - mu )/( sigma/sqrt(n) )
c = qnorm( alpha ) 

mu_0 = 48 # this is the mean we are trying to refute
mu = 46 # the true mean is this
pnorm( c - ( sqrt(n)*(mu-mu_0)/sigma ) )

ns = c(20,30,40)
pnorm( c - ( sqrt(ns)*(mu-mu_0)/sigma ) )

# Problem 20:
#
alpha = 0.05
n = 25
mu_0= 42

c = qt(1-0.5*alpha,n-1) # 2.063899

# Part a:
#
x_bar = 44
s = 10
T = ( x_bar - mu_0 ) / ( s/sqrt(n) )
if( abs(T)>c ){
  print("reject H_0");
}else{
  print("cannot reject H_0");
}

# Part b:
#
x_bar = 43
s = 10
T = ( x_bar - mu_0 ) / ( s/sqrt(n) )
if( abs(T)>c ){
  print("reject H_0");
}else{
  print("cannot reject H_0");
}

# Part c:
#
x_bar = 43
s = 2
T = ( x_bar - mu_0 ) / ( s/sqrt(n) )
if( abs(T)>c ){
  print("reject H_0");
}else{
  print("cannot reject H_0");
}

# Problem 22:
#
alpha = 0.05
n = 16
mu_0= 42

c = qt(1-alpha,n-1) 

# Part (a):
#
x_bar = 44
s = 10
T = ( x_bar - mu_0 )/( s/sqrt(n) )

if( T > c ){
  print("we reject H_0");
}else{
  print("we cannot reject H_0")
}

# Part (b):
#
x_bar = 43
s = 10
T = ( x_bar - mu_0 )/( s/sqrt(n) )

if( T > c ){
  print("we reject H_0");
}else{
  print("we cannot reject H_0")
}

# Part (c):
#
x_bar = 43
s = 2
T = ( x_bar - mu_0 )/( s/sqrt(n) )

if( T > c ){
  print("we reject H_0");
}else{
  print("we cannot reject H_0")
}


# Problem 23:
#
alpha = 0.05
n = 16
mu_0= 42

c = qt(alpha,n-1) 

# Part (a):
#
x_bar = 44
s = 10
T = ( x_bar - mu_0 )/( s/sqrt(n) )

if( T < c ){
  print("we reject H_0");
}else{
  print("we cannot reject H_0")
}

# Part (b):
#
x_bar = 43
s = 10
T = ( x_bar - mu_0 )/( s/sqrt(n) )

if( T < c ){
  print("we reject H_0");
}else{
  print("we cannot reject H_0")
}

# Part (c):
#
x_bar = 43
s = 2
T = ( x_bar - mu_0 )/( s/sqrt(n) )

if( T < c ){
  print("we reject H_0");
}else{
  print("we cannot reject H_0")
}


# Problem 24:
#
data = c(38, 44, 62, 72, 43, 40, 43, 42, 39, 41)
xbar = mean(data)
s = sqrt(var(data))
n = length(data)

alpha = 0.05
c = qt( 1 - 0.5*alpha, n-1 )

mu_0 = 45
T = ( xbar - mu_0)/( s/sqrt(n) )
if( abs(T)>c ){
  print("we reject H_0")
}else{
  print("we cannot reject H_0")
}

# Problem 25:
#
xbar = 9.79
s = 2.71
n = 100

alpha = 0.025
c = qt( alpha, n-1 )

mu_0 = 10.5
T = ( xbar - mu_0)/( s/sqrt(n) )
if( T<c ){
  print("we reject H_0")
}else{
  print("we cannot reject H_0")
}

# Problem 26:
#
xbar = 40
s = 4
n = 16

alpha = 0.01
c = qt( 1-alpha, n-1 )

mu_0 = 38
T = ( xbar - mu_0 )/( s/sqrt(n) )
if( T>c ){
  print("we reject H_0")
}else{
  print("we cannot reject H_0")
}


# Problem 27:
# I think there is a typo in this problem
# 
xbar = 76
s = 4
n = 9

alpha = 0.05
c = qt( 1-0.5alpha, n-1 )

mu_0 = 72
T = ( xbar - mu_0 )/( s/sqrt(n) )
if( abs(T)>c ){
  print("we reject H_0")
}else{
  print("we cannot reject H_0")
}

# Problem 28:
#
xbar = 146
s = 2.5
n = 10

alpha = 0.05
c = qt( 1-0.5*alpha, n-1 )

mu_0 = 150
T = ( xbar - mu_0 )/( s/sqrt(n) )
if( abs(T)>c ){
  print("we reject H_0")
}else{
  print("we cannot reject H_0")
}

# Problem 29:
#
data = c(42,90,84,87,116,95,86,99,93,92,121,71,66,98,79,102,60,112,105,98)
xbar = mean(data)
s = sqrt(var(data))
n = length(data)

alpha = 0.01
c = qt( 1-alpha, n-1 )

mu_0 = 80
T = ( xbar - mu_0 )/( s/sqrt(n) )
if( T>c ){
  print("we reject H_0")
}else{
  print("we cannot reject H_0")
}


# Problem 30:
#
xTrimmedMean = 44
xTrimmedMean = 43
sw = 9

mu_t = 42
alpha = 0.05
n = 20

# get the amount of trimming h:
h = 2*floor(0.2*n)

c = qt(1-0.5*alpha,n-h-1)

T_t = 0.6*(xTrimmedMean - mu_t)/(sw/sqrt(n))


# Problem 31
#
#Part (a):
xTrimmedMean = 44
sw = 9

mu_t = 42
alpha = 0.05
n = 16

# get the amount of trimming h:
h = 2*floor(0.2*n)

T_t = 0.6*(xTrimmedMean - mu_t)/(sw/sqrt(n))

c = qt(1-alpha,n-h-1)

#Part (b):
xTrimmedMean = 43
T_t = 0.6*(xTrimmedMean - mu_t)/(sw/sqrt(n))

#Part (c):
xTrimmedMean = 43
sw = 3
T_t = 0.6*(xTrimmedMean - mu_t)/(sw/sqrt(n))


# Problem 32
#
xTrimmedMean = 42.17
sw = 1.73
n = 10
mu_t = 45
T_t = 0.6*(xTrimmedMean - mu_t)/(sw/sqrt(n))
h = 2 * floor(.2*n)
alpha = 0.05
c = qt( 1 - 0.5*alpha, n - h - 1 )

# Problem 33
#
mu_t = 4.8
xTrimmedMean = 5.1
sw = 7
n=25
alpha = 0.01
h = 2*floor(0.2*n)

c = qt( 1 - 0.5*alpha, n - h - 1 )

T_t = 0.6*( xTrimmedMean - mu_t )/( sw/sqrt(n) )