# Section 2; Question 1 (EPage 661):
#
source('../Chapter8/chap_8_sect_2_question_9_data.R')
n = dim(DF)[1]
mu0_tilde = 9
k = sum( DF$Number > mu0_tilde )
p_value = sum( c( dbinom( 0:(n-k), n, 0.5 ), dbinom( k:n, n, 0.5 ) ) )
print( sprintf( 'k= %d; p_value= %10.6f', k, p_value ) )


# Section 2; Question 2 (EPage 662):
#
source('../Chapter8/chap_8_sect_2_question_12_data.R')
n = dim(DF)[1]
mu0_tilde = 0.12
k = sum( DF$Chirp_Length > mu0_tilde )
exact_prob = sum( dbinom( 0:k, n, 0.5 ) )
z = ( k - n/2 )/sqrt( n/4 )
norm_approx = pnorm( z )
print( sprintf( 'exact_prob= %.6f; norm_approx= %.6f', exact_prob, norm_approx ) )


# Section 2; Question 3:
#
source('chap_14_sect_2_question_3_data.R')

mu0_tilde = log(2)
n = length(data)
k = sum( data > mu0_tilde )
z = ( k - n/2 )/sqrt( n/4 )
p_value = pnorm(z) + ( 1 - pnorm(-z) )
print( sprintf( 'p_value= %.6f', p_value ) )

# Section 2; Question 4:
#
n = 22
sigma2 = 6.0
z_alpha = -qnorm( 0.05 )
x_crit = sqrt(n/4) * z_alpha + n/2
p_alt = 1 - pnorm( 10, mean=11, sd=sqrt(6) )
1 - pbinom( floor(x_crit), n, p_alt )


# Section 2; Question 5:
#
n = 7
dbinom( 0:n, n, 1/2 )
ts = 0:n
alphas = 1 - pbinom( ts, n, 1/2 )

# check vs. the back of the book:
db = c( 1, 7, 21, 35, 35, 21, 7, 1 )/128
cumsum( db )


# Section 2; Question 6:
#
source('../Chapter7/case_study_7_4_2_data.R')

golden_ratio = ( sqrt(5) - 1 ) / 2

k = sum( data > golden_ratio )
n = length(data)
z = ( k - n/2 ) / sqrt( n/4 )
p_value = pnorm( -z ) + ( 1 - pnorm( z ) ) 
print( sprintf( 'p_value= %.6f', p_value ) )


# Section 2; Question 7:
#
source('../Chapter5/chap_5_sect_3_question_2_data.R')

n = dim(DF)[1]
k = sum( DF$Ratio > 0.8 )
z = ( k - n/2 ) / sqrt( n/4 )
p_value = pnorm( z )
print( sprintf( 'large sample sign test p_value= %.6f', p_value ) )
print( t.test( DF$Ratio, mu=0.8, alternative=c("less"), conf.level=0.90 ) )


# Section 2; Question 8:
#
source('../Chapter13/chap_13_sect_2_question_1_data.R')

k = sum( DF$hypnotic > DF$waking )
n = dim(DF)[1]
z = ( k - n/2 )/sqrt( n/4 )
p_value = 1 - pnorm( z )
print( sprintf( 'p_value= %.6f', p_value ) )


# Section 2; Question 9:
n = 28
k = 19
z = ( k - n/2 )/sqrt( n/4 )
p_value = pnorm( -z ) + ( 1 - pnorm( z ) )
print( sprintf( 'p_value= %.6f', p_value ) )


# Section 2; Question 10:
#
t = 6
theta = t/0.25

n = 36
n * 0.25
n * 0.25 * (1-0.25)

y_plus_values = 0:n
prob_y_plus_values = dbinom( y_plus_values, n, 22/28 )
z_values = ( y_plus_values - 9 ) / sqrt( 6.75 )
z_alpha = -qnorm( 0.05 )
dont_reject_null = abs( z_values ) < z_alpha
prob_type_II_error = sum( prob_y_plus_values[dont_reject_null] )
print( sprintf( 'p_value= %.6f', prob_type_II_error ) )


# Section 2; Question 11:
#
source('../Chapter13/case_study_13_3_1_data.R')

n = dim(DF)[1]
k = sum( DF$After > DF$Before )
p_value = sum( c( dbinom( 0:k, n, 0.5 ), dbinom( (n-k):n, n, 0.5 ) ) )
print( sprintf( 'k= %d; p_value= %.6f', k, p_value ) )