# Section 4; Question 1:
x = 55
n = 200
p_x = x/n
y = 40
m = 200
p_y = y/m

p_e = (x+y)/(n+m)

z = ( p_x - p_y )/sqrt( p_e*(1-p_e)/n + p_e*(1-p_e)/m )
p_value = 2*(1-pnorm(abs(z))) # two-sided test

alpha = 0.05
z_crit = qnorm( 1-alpha/2 )
print(sprintf('z= %.6f; p_value= %.6f; z_crit= %.6f', z, p_value, z_crit))


# Section 4; Question 2:
#
x = 66 # the restricted diet group 
n = 423
p_x = x/n
y = 93 # the control group
m = 423
p_y = y/m

p_e = (x+y)/(n+m)

z = ( p_x - p_y )/sqrt( p_e*(1-p_e)/n + p_e*(1-p_e)/m )
p_value = 2*(1-pnorm(abs(z))) # two-sided test

alpha = 0.05
z_crit = qnorm( 1-alpha/2 )
print(sprintf('z= %.6f; p_value= %.6f; z_crit= %.6f', z, p_value, z_crit))


# Section 4; Question 3:
#
x = 24 # witched wells
n = 29
p_x = x/n
y = 27 # non witched wells
m = 32
p_y = y/m

p_e = (x+y)/(n+m)

z = ( p_x - p_y )/sqrt( p_e*(1-p_e)/n + p_e*(1-p_e)/m )
p_value = 2*(1-pnorm(abs(z))) # two-sided test

alpha = 0.05
z_crit = qnorm( 1-alpha/2 )
print(sprintf('z= %.6f; p_value= %.6f; z_crit= %.6f', z, p_value, z_crit))


# Section 4; Question 4:
#
x = 53 # in spain
n = 53+38
p_x = x/n
y = 705 # not in spain
m = 705+412
p_y = y/m

p_e = (x+y)/(n+m)

z = ( p_x - p_y )/sqrt( p_e*(1-p_e)/n + p_e*(1-p_e)/m )
p_value = 2*(1-pnorm(abs(z))) # two-sided test

alpha = 0.01
z_crit = qnorm( 1-alpha/2 )
print(sprintf('z= %.6f; p_value= %.6f; z_crit= %.6f', z, p_value, z_crit))


# Section 4; Question 5:
#
p_x = 0.617
n = 1675
x = n*p_x

p_y = 0.521
m = 660
y = m*p_y

p_e = (x+y)/(n+m)

z = ( p_x - p_y )/sqrt( p_e*(1-p_e)/n + p_e*(1-p_e)/m )
p_value = 1-pnorm(z) # one-sided test

alpha = 0.01
z_crit = qnorm( 1-alpha )
print(sprintf('z= % 6f; p_value= %.6f; z_crit= %.6f', z, p_value, z_crit))


# Section 4; Question 6:
#
x = 60
n = 100
p_x = x/n
y = 48
m = 100
p_y = y/m

p_e = (x+y)/(n+m)

z = ( p_x - p_y )/sqrt( p_e*(1-p_e)/n + p_e*(1-p_e)/m )
p_value = 2*(1-pnorm(abs(z))) # two-sided test
print(sprintf('z= %.6f; p_value= %.6f', z, p_value))


# Section 4; Question 7:
#
x = 2915
n = 6001
p_x = x/n
y = 1219
m = 2604
p_y = y/m

p_e = (x+y)/(n+m)

z = ( p_x - p_y )/sqrt( p_e*(1-p_e)/n + p_e*(1-p_e)/m )
p_value = 2*(1-pnorm(abs(z))) # two-sided test

alpha = 0.05
z_crit = qnorm( 1-alpha/2 )
print(sprintf('z= %.6f; p_value= %.6f; z_crit= %.6f', z, p_value, z_crit))


# Section 4; Question 8:
#
x = 175
n = 609
p_x = x/n
y = 100
m = 160
p_y = y/m

p_e = (x+y)/(n+m)

z = ( p_x - p_y )/sqrt( p_e*(1-p_e)/n + p_e*(1-p_e)/m )
p_value = 2*(1-pnorm(abs(z))) # two-sided test

alpha = 0.05
z_crit = qnorm( 1-alpha/2 )
print(sprintf('z= %.6f; p_value= %.6f; z_crit= %.6f', z, p_value, z_crit))


# Section 4; Question 9:
#
p_x = 0.26
n = 300
x = p_x*n
p_y = 0.25
m = 200
y = p_y*m

p_e = (x+y)/(n+m)

z = ( p_x - p_y )/sqrt( p_e*(1-p_e)/n + p_e*(1-p_e)/m )
p_value = 2*(1-pnorm(abs(z))) # two-sided test

alpha = 0.05
z_crit = qnorm( 1-alpha/2 )
print(sprintf('z= %.6f; p_value= %.6f; z_crit= %.6f', z, p_value, z_crit))


# Section 4: Question 10:
#
x = 55 # men
n = 160
p_x = x/n
y = 60 # women
m = 192
p_y = y/m

p_e = (x+y)/(n+m)

numer = p_e^(x+y) * ( 1 - p_e )^(n+m-x-y)
denom = p_x^x * ( 1 - p_x )^(n-x) * p_y^y * ( 1 - p_y )^(m-y)
lam = numer/denom
print(-2*log(lam))