# Section 3; Question 1:
#
n = 52
p0 = 0.4
ll = n*p0 - 3*sqrt(n*p0*(1-p0))
ul = n*p0 + 3*sqrt(n*p0*(1-p0))
print( c(0, ll, ul, n) ) # make sure we can use the large sample tests
z = (24 - n*p0) / sqrt( n*p0*(1-p0) )
z_alpha = qnorm(1-0.05)
p_value = 1-pnorm(z)
print(sprintf('z= %f; z_alpha= %f; P-value= %f', z, z_alpha, p_value))


# Section 3; Question 2:
#
p0 = 0.67
n = 35
ll = n*p0 - 3*sqrt(n*p0*(1-p0))
ul = n*p0 + 3*sqrt(n*p0*(1-p0))
print( c(0, ll, ul, n) )
z = (18 - n*p0) / sqrt( n*p0*(1-p0) )
z_alpha = qnorm(1-0.05/2)
p_value = 2*(1-pnorm(abs(z)))
print(sprintf('z= %f; z_alpha/2= %f; P-value= %f', z, z_alpha, p_value))


# Section 3; Question 3:
#
p0 = 0.65
n = 120
ll = n*p0 - 3*sqrt(n*p0*(1-p0))
ul = n*p0 + 3*sqrt(n*p0*(1-p0))
print( c(0, ll, ul, n) )
z = (72 - n*p0) / sqrt( n*p0*(1-p0) )
z_alpha = qnorm(1-0.05)
p_value = pnorm(z)
print(sprintf('z= %f; -z_alpha= %f; P-value= %f', z, -z_alpha, p_value))


# Section 3; Question 4:
#
p0 = 0.45
n = 200
ll = n*p0 - 3*sqrt(n*p0*(1-p0))
ul = n*p0 + 3*sqrt(n*p0*(1-p0))
print( c(0, ll, ul, n) )
num_of_successes = 90:110 # n*p0 = 90 so when n*p0 = 90 our z score is zero
z = (num_of_successes - n*p0) / sqrt( n*p0*(1-p0) )
p_value = 1-pnorm(z)

inds = p_value<=0.14
print('p-values:')
print(p_value[inds])
print('Number of successes:')
print(num_of_successes[inds])


# Section 3; Question 5:
#
source('../Chapter5/table_5_3_3.R')
m = -log(0.5)
n = length(data)
p0 = 0.5
ll = n*p0 - 3*sqrt(n*p0*(1-p0))
ul = n*p0 + 3*sqrt(n*p0*(1-p0))
print( c(0, ll, ul, n) )

x = sum( data < m )
p_hat = x/n
print(sprintf('x= %d; n= %d; p0= %f; p_hat= %f', x, n, p0, p_hat))
z = (x - n*p0) / sqrt( n*p0*(1-p0) )
p_value = 2*(1-pnorm(abs(z))) # for a two sided test
print(sprintf('z= %f; P-value= %f', z, p_value))


# Section 3; Question 6:
#
p0 = 1/12
n = 348
ll = n*p0 - 3*sqrt(n*p0*(1-p0))
ul = n*p0 + 3*sqrt(n*p0*(1-p0))
print( c(0, ll, ul, n) )

x = 16
p_hat = x/n
print(sprintf('x= %d; n= %d; p0= %f; p_hat= %f', x, n, p0, p_hat))
z = (x - n*p0) / sqrt( n*p0*(1-p0) )
p_value = 2*(1-pnorm(abs(z))) # for a two sided test
print(sprintf('z= %f; P-value= %f', z, p_value))


# Section 3; Question 7:
#
k_star = 4:7
d = dbinom( k_star, 7, 0.5 )
print(rev(cumsum(rev(d))))


# Section 3; Question 9:
#
alpha = sum( dbinom( 0:3, 7, 0.75 ) )
print(alpha)
ps = seq( 0.5, 0.9, length.out=100 )
powers = rep(NA, length(ps))
for(ii in 1:length(ps)){
    powers[ii] = 1 - sum( dbinom( 0:3, 7, ps[ii] ) )
}
plot( ps, powers, type='l', xlab='p', ylab='power' )
grid()