# Question 1:
#
pt_a = qchisq( 0.95, 14 )
pt_b = qchisq( 0.9, 2 )
pt_c = qchisq( 0.025, 9 )
print( c( pt_a, pt_b, pt_c ) )


# Question 2:
#
pt_a = 1 - pchisq( 8.672, 17 )
pt_b = pchisq( 10.645, 6 )
pt_c = pchisq( 34.17, 20 ) - pchisq( 9.591, 20 )
pt_d = pchisq( 9.21, 2 )
print( c( pt_a, pt_b, pt_c, pt_d ) )


# Question 3:
#
pt_a = qchisq( 1-0.99, 9 )
pt_b = qchisq( 0.05, 15 )
pt_c = qchisq( 0.09, 22 ) + pchisq( 9.542, 22 )
pt_d = qchisq( pchisq( 48.232, 31 ) - 0.95, 31 )
print( c( pt_a, pt_b, pt_c, pt_d ) )


# Question 4:
#
library(stats)

qchi2_root = function(x){
    pchisq(5.009, x) - 0.0125
}
res = uniroot( qchi2_root, c(10, 20) )
pt_a = res$root

qchi2_root = function(x){
    pchisq(30.144, x) - ( 0.05 - pchisq(27.204, x) )
}
res = uniroot( qchi2_root, c(40, 60) )
pt_b = res$root

qchi2_root = function(x){
    pchisq(19.281, x) - 0.05
}
res = uniroot( qchi2_root, c(20, 40) )
pt_c = res$root

qchi2_root = function(x){
    pchisq(24.769, x) - ( 0.80 - pchisq(10.085, x) )
}
res = uniroot( qchi2_root, c(20, 30) )
pt_d = res$root


print( c( pt_a, pt_b, pt_c, pt_d ) ) 


# Question 5:
#
n = 200
z_p = qnorm(0.95)
print( n * ( 1 - 2/(9*n) + z_p * sqrt( 2 / (9*n) ) )^3 )


# Question 6:
#
n=2
v = pchisq( 2*(n-1), n-1 ) 
while( v < 0.95 ){
    n = n+1
    v = pchisq( 2*(n-1), n-1 )
}
print( c( n, v ) )


# Question 8:
#
n = 19
sigma2 = 12
alpha = 0.05
print( ( sigma2 / (n-1) ) * c( qchisq(alpha/2, n-1), qchisq(1-alpha/2, n-1) ) )


# Question 9:
#
source('chap_7_sect_5_question_9_data.R')
n = length(data)
s2 = var(data)
alpha = 0.05
print( sqrt( ( (n-1) * s2 ) / qchisq( c(1-alpha/2, alpha/2), n-1 ) ) )

lower_limit = sqrt( ( (n-1) * s2 ) / qchisq( 1-alpha, n-1 ) )
upper_limit = sqrt( ( (n-1) * s2 ) / qchisq( alpha, n-1 ) )
print( c( lower_limit, upper_limit ) ) 


# Question 10:
#
source('chap_7_sect_5_question_10_data.R')
data = DF$Interest_Rate
s2 = var(data)
n = length(data)
alpha = 0.05
ci = sqrt( ( (n-1) * s2 ) / qchisq( c(1-alpha/2, alpha/2), n-1 ) )
print(ci)


# Question 12:
#
source('case_study_7_5_1_data.R')
n = length(data)
s2 = var(data)
alpha = 0.05
z_crit = qnorm(1-alpha/2)
ci = sqrt( (n-1) * s2 * c( 1/(n-1 + sqrt(2*(n-1)) * z_crit), 1/(n-1 - sqrt(2*(n-1)) * z_crit) ) )
print( ci )


# Question 13:
#
library(stats)

qchi2_root = function(x){
    qchisq(0.95, x-1) / qchisq(0.05, x-1) - 261.9/54.47
}
res = uniroot( qchi2_root, c(10, 20) )
print( res$root )
n = 11 # this is closest integer value
s2 = ( 54.47 * qchisq(1-alpha/2, n-1) )/(n-1)
print(s2)


# Question 15:
#
source('case_study_7_5_1_data.R')
n = length(data) 
s2 = var(data)
sigma0 = 30.4
c2 = (n-1)*s2/sigma0^2
print(sprintf('chi^2= %f; p_value= %f', c2, pchisq(c2, n-1))) 


# Question 16:
#
source('chap_7_sect_5_question_16_data.R')
n = length(data)
s2 = var(data)
sigma0 = 1
c2 = (n-1)*s2/sigma0^2
print(sprintf('s^2= %f; chi^2= %f; p_value= %f', s2, c2, 1-pchisq(c2, n-1))) 


# Question 17:
#
n = 24
ybar = 0.115
s = 0.1017
mu0 = 0.101
sigma0 = 0.1567
t = ( ybar - mu0 )/(s/sqrt(n))
p_value = 1-pt(abs(t), n-1)
print(sprintf('t= %f; p_value= %f', t, p_value))

c2 = (n-1)*s^2/sigma0^2
print(sprintf('chi^2= %f; p_value= %f', c2, pchisq(c2, n-1)))