#
# 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.
#
#-----

# Ex 7.47:
#
DF = read.csv( "../../Data/CH07/ex7-47.txt", header=TRUE, quote="'" )
X = DF$strength

n = length(X)
xbar = mean(X)
s = sd(X)
z_crit = qnorm( 1 - 0.05/2 )

xbar + ( s / sqrt(n) ) * z_crit * c(-1,+1)

p_hat = sum( X > 10 ) / length(X)
ci_population_proportion( p_hat, n, 0.05 )



# Ex 7.49:
#
DF = read.csv( "../../Data/CH07/ex7-49.txt", header=TRUE, quote="'" )
X = DF$Volume..

boxplot( X )

qqnorm( X )
qqline( X )
grid ()

n = length(X)
m = mean(X)
s = sd(X)
t_crit = qt( 1 - 0.02/2, n-1 )
m + ( s / sqrt(n) ) * t_crit * c(-1,+1)

# Ex 7.50:
#
ci = c( 229.764, 233.504 )
m = 0.5 * sum( ci )

t_crit = qt( 1 - 0.05/2, n-1 )

ci_std_partial = t_crit / sqrt(n)
s = diff( ci ) / ( 2 * ci_std_partial )

t_crit = qt( 1 - 0.01/2, n-1 ) # the new CI critical value
m + ( s / sqrt(n) ) * t_crit * c(-1,+1)



# 7.51:
#
p_hat = 136/200
ci_population_proportion( p_hat, 200, 0.1 )

q_hat = 1 - p_hat
w = 0.05
z_crit = qnorm( 1-0.1/2 )
( 4 * z_crit^2 * p_hat * q_hat ) / ( w^2 )

# Ex 7.52
#
n = 5
m = 24.3
s = 4.1
t_crit = qt( 1 - 0.05/2, n-1 )

m + ( s / sqrt(n) ) * t_crit * c(-1,+1)

alpha = 0.1
sqrt( (n-1) * s^2 / qchisq( alpha, n-1 ) )

s_prediction = s * sqrt( 1 + 1/n )
t_crit = qt( 1 - 0.05/2, n-1 )
m + s_prediction * t_crit * c(-1,+1)

# Ex 7.53:
#
theta_hat = (1/3) * ( 10.5 + 10.0 + 10.1 ) - 10.7
var_theta_hat = (1/9) * ( 1.5^2 / 100 + 1.3^2 / 90 + 1.8^2 / 100 ) + 1.6^2 / 120

alpha = 0.05
z_crit = qnorm( 1 - alpha/2 )

theta_hat + sqrt( var_theta_hat ) * z_crit * c(-1,+1)


# Ex 7.54:
#
p_hat = 11/55

ci_population_proportion( p_hat, 55, 0.10 )


# Ex 7.55:
#
s = 0.8
z_crit = qnorm( 1 - 0.05/2 )
w = 0.2
( 2*s*z_crit / w )^2


# Ex 7.56:
#
DF = read.csv( "../../Data/CH07/ex7-56.txt", header=TRUE, quote="'" )
X = DF$compliance

qqnorm( X )
qqline( X )
grid()

n = length(X)
m = mean(X)
s = sd(X)
t_crit = qt( 1 - 0.05/2, n-1 )

m + ( s / sqrt(n) ) * t_crit * c(-1,+1)


# Ex 7.57:
#
n = 20
data = c( 11, 15, 29, 33, 35, 40, 47, 55, 58, 72 )
r = length( data )
tr = sum( data ) + (n-r) * data[length(data)]

alpha = 0.05
2 * tr / c( qchisq( 1 - alpha/2, 2*r ), qchisq( alpha/2, 2*r ) )


# Ex 7.58:
#
DF = read.csv( "../../Data/CH07/ex7-58.txt", header=TRUE, quote="'" )
X = DF$alanine
n = length(X)
range( X )


# Ex 7.59:
#
X = c( 4.2, 3.5, 1.7, 1.2, 2.4 )
n = length(X)
y = max(X)

alpha = 0.05
alpha_over_two = alpha / 2
ci_1 = c( y / ( ( 1 - alpha_over_two )^(1/n) ), y / ( alpha_over_two^(1/n) ) )

ci_2 = c( y, y / alpha^(1/n) )


# Ex 7.61:
#
DF = read.csv( "../../Data/CH07/ex7-45.txt", header=TRUE, quote="'" )
X = DF$toughness

n = length(X)

# the suggested robust CI:
#
conservative_t_crit = 1.94

# Compute the fourth spread:
#
tilde_mu = median(X)

lower_fourth = median( X[ X<=tilde_mu ] )
upper_fourth = median( X[ X>=tilde_mu ] )
fs = upper_fourth - lower_fourth

tilde_mu + ( conservative_t_crit / 1.075 ) * ( fs / sqrt(n) ) * c(-1,+1)

# a standard t-based CI:
#
s = sd(X)
alpha = 0.05
t_crit = qt( 1 - alpha/2, n-1 )
mean(X) + ( s / sqrt(n) ) * t_crit * c(-1,+1)


# Ex 7.62:
#
X = c( 41.53, 18.73, 2.99, 30.34, 12.33, 117.52, 73.02, 223.63, 4.00, 26.78 )
n = length(X)

alpha = 0.05
chisq_crit = qchisq( alpha, 2*n )

chisq_crit / ( 2*sum(X) )

chisq_crit = qchisq( 1-alpha, 2*n )
lamb = chisq_crit / ( 2*sum(X) )

exp( -lamb * (100) )