# Section 3; Question 1:
#
sigma = 15
n = 50
y_bar = 107.9
z_crit = qnorm( 1 - 0.05/2 )
print( y_bar + sigma/sqrt(n) * z_crit * c(-1,+1) )


# Section 3; Question 2:
#
source('chap_5_sect_3_question_2_data.R')
x_bar = mean( DF$Ratio )
n = dim(DF)[1]
sigma = 0.09 
z_crit = qnorm( 1 - 0.05/2 )
print( x_bar + sigma/sqrt(n) * z_crit * c(-1,+1) )


# Section 3: Question 3:
#
data = c( 52, 69, 73, 88, 87, 56 ) 
sigma = 8
n = length(data)
x_bar = mean(data)
z_crit = qnorm( 1 - 0.05/2 )
print( x_bar + sigma/sqrt(n) * z_crit * c(-1,+1) )


# Section 3: Question 4:
#
n = 38
x_bar = 188.4
sigma = 40.7
z_crit = qnorm( 1 - 0.05/2 )
print( x_bar + sigma/sqrt(n) * z_crit * c(-1,+1) )


# Section 3: Question 5:
#
sigma = 14.3
alpha = 0.05
z_crit = qnorm( 1 - alpha/2 )
print( ( 2 * z_crit * sigma / 3.06 )^2 )


# Section 3: Question 6:
#
z_left = c( -1.64, -10, -1.64 )
z_right = c( 2.33, 2.58, 0 )
print( round( pnorm( z_right ) - pnorm( z_left ), 2 ) )


# Section 3: Question 7:
#
p = pnorm( 1.06 ) - pnorm( -0.96 )
print( sum( dbinom( 4:5, 5, p ) ) ) 


# Section 3: Question 9:
#
data = c( 2.5, 3.2, 0.5, 0.4, 0.3, 0.1, 0.1, 0.2, 7.4, 8.6, 0.2, 0.1, 0.4, 1.8, 0.3, 1.3, 1.4, 11.2, 2.1, 10.1 )
print( c( length(data), mean(data) ) )
hist( data ) # note that this data is not normal looking 


# Section 3: Question 10:
#
n = 540
p_hat = 192/n
alpha = 0.05
z_crit = qnorm( 1 - alpha/2 )
s = sqrt( p_hat * ( 1 - p_hat ) / n )
print( p_hat + z_crit * s * c( -1, +1 ) )


# Section 3: Question 11:
#
n = 1015
p_hat = 281/n
alpha = 0.1
z_crit = qnorm( 1 - alpha/2 )
s = sqrt( p_hat * ( 1 - p_hat ) / n )
print( p_hat + z_crit * s * c( -1, +1 ) )


# Section 3: Question 12:
#
n = 100
p_hat = 54/n
alpha = 0.05
z_crit = qnorm( 1 - alpha/2 )
s = sqrt( p_hat * ( 1 - p_hat ) / n )
print( p_hat + z_crit * s * c( -1, +1 ) )


# Section 3: Question 13:
#
n = 2253
p_hat = 0.63
alpha = 0.05
z_crit = qnorm( 1 - alpha/2 )
s = sqrt( p_hat * ( 1 - p_hat ) / n )
print( p_hat + z_crit * s * c( -1, +1 ) )


# Section 3: Question 14:
#
int = c( 0.57, 0.63 )
p_hat = mean( int )
print( p_hat )

alpha = 1-0.5
z_crit = qnorm( 1 - alpha/2 )
print( z_crit ) 
print( 1 / ( (diff( int )/(2*z_crit))^2 / (p_hat * (1-p_hat)) ) )


# Section 3: Question 15:
#
alpha = 0.01
z_crit = qnorm( 1 - alpha/2 )
print( 1 / ( 4 * ( 0.02/(2*z_crit) )^2 ) )


# Section 3: Question 16:
#
n = 14000
ps = seq( 0.52, 0.6, length.out=500 )
lhs = rep( NA, length(ps) )
for( ii in 1:length(ps) ){
    lhs[ii] = sum( dbinom( 8088:14000, n, ps[ii] ) )
}
plot( ps, lhs, type='l' )
abline(h=0.2, col='red')
grid()

mi = which.min(abs(lhs-0.2))
print( c(lhs[mi], ps[mi]) )


# Section 3: Question 17:
#
print( pnorm( 0.67 ) - pnorm( -0.67 ) ) # area covered by the first interval 
print( 1 - pnorm( 0 ) ) # area covered by the second interval 


# Section 3: Question 19:
#
n = 998
p_hat = 0.59
d = 1.96/(2*sqrt(n))
print(sprintf('d= %f', d))
alpha = 0.05
z_crit = qnorm( 1 - alpha/2 )
s = sqrt( p_hat * ( 1 - p_hat ) / n )
print( p_hat + z_crit * s * c( -1, +1 ) )


# Section 3: Question 20:
#
n = 202
p_hat = 86/n
d = 1.96/(2*sqrt(n))
print(sprintf('p_hat= %f; d= %f', p_hat, d))


# Section 3: Question 22:
#
n = 350
p_hat = 126/n
alpha = 0.1
z_crit = qnorm( 1 - alpha/2 )
s = sqrt( p_hat * ( 1 - p_hat ) / n )
ci = p_hat + z_crit * s * c( -1, +1 )
print(sprintf('ci Part (a): (%f, %f)', ci[1], ci[2]))
N = 3000
s = sqrt( p_hat * ( 1 - p_hat ) / n ) * sqrt( (N-n)/(N-1) )
ci = p_hat + z_crit * s * c( -1, +1 )
print(sprintf('ci Part (b): (%f, %f)', ci[1], ci[2]))


# Section 3: Question 23:
#
d = 0.06
n = 1/( (2*(d/1.96))^2 )
print(n)
print(4*n)


# Section 3: Question 25:
#
alpha_1 = 0.04
d_1 = 0.05
z_crit = qnorm( 1 - alpha_1/2 )
n_1 = z_crit^2 / (4*d_1^2)

alpha_2 = 0.08
d_2 = 0.04
z_crit = qnorm( 1 - alpha_2/2 )
n_2 = z_crit^2 / (4*d_2^2)

print( c( n_1, n_2 ) )


# Section 3: Question 26:
#
B = 0.4*0.6
alpha = 0.01
z_crit = qnorm( 1 - alpha/2 )
d = 0.05
print( ((z_crit^2)*B)/(d^2) )


# Section 3: Question 27:
#
alpha = 0.2
d = 0.02
z_crit = qnorm( 1 - alpha/2 )
n = z_crit^2 / (4*d^2)
print( n )


# Section 3: Question 28:
#
alpha = 0.15
d = 0.03
z_crit = qnorm( 1 - alpha/2 )
n = z_crit^2 / (4*d^2)
print( n )
B = 0.09 # using the known bound on B we get
n = ((z_crit^2)*B)/ d^2
print( n )