# Section 2; Question 1:
#
z = (114.2 - 120)/(18/sqrt(25))
z_alpha = qnorm(1-0.08)
p_value = pnorm(z)
print(sprintf('z= %f; -z_alpha= %f; p_value= %f', z, -z_alpha, p_value))

z = (45.1 - 42.9)/(3.2/sqrt(16))
z_alpha = qnorm(1-0.01/2)
p_value = 2*(1-pnorm(abs(z)))
print(sprintf('z= %f; z_alpha/2= %f; p_value= %f', z, z_alpha, p_value))

z = (15.8 - 14.2)/(4.1/sqrt(9))
z_alpha = qnorm(1-0.13)
p_value = 1-pnorm(z)
print(sprintf('z= %f; z_alpha= %f; p_value= %f', z, z_alpha, p_value))


# Section 2; Question 2:
#
z_crit = qnorm(1-0.06/2)
y_limits = 95 + (15/sqrt(22)) * z_crit * c(-1, +1)
print(y_limits)


# Section 2; Question 4:
#
z = (33800 - 32500)/(4000/sqrt(15))
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 2; Question 6:
#
cr = c( 29.9, 30.1 )
zr = ( cr - 30 )/(6/sqrt(16))
alpha = pnorm(zr[2])-pnorm(zr[1])
print(zr)
print(sprintf('level of significance (alpha)= %f', alpha))
z_crit = qnorm(1-alpha/2)
y_limits = 30 + (6/sqrt(16)) * z_crit * c(-1, +1)
print(sprintf('reject H_0 if y < %f or y > %f', y_limits[1], y_limits[2]))


# Section 2; Question 7:
#
data = c( 12.3, 12.7, 13.6, 12.7, 12.9, 12.6,
	  12.6, 13.1, 12.6, 13.1, 12.7, 12.5,
	  13.2, 12.8, 12.4, 12.6, 12.4, 12.4,
	  13.1, 12.9, 13.3, 12.6, 12.6, 12.7,
	  13.1, 12.4, 12.4, 13.1, 12.4, 12.9 )
n = length(data)
ybar = mean(data)
s = 0.4
z = (ybar - 12.6)/(s/sqrt(n))
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))
hist(data)


# Section 2; Question 9:
#
mu0 = 120
ybar = 122.3
sigma = 10
n = 16
z = (ybar - mu0)/(sigma/sqrt(n))
p_value = 2*(1-pnorm(abs(z)))
print(sprintf('z= %f; p_value= %f', z, p_value))


# Section 2; Question 10:
#
mu0 = 120
sigma = 12
n = 50
ybar = 125.2
z = (ybar - mu0)/(sigma/sqrt(n))
p_value = 1-pnorm(z)
print(sprintf('z= %f; p_value= %f', z, p_value))


# Section 2; Question 11:
#
mu0 = 145.75
sigma = 9.5
n = 25
ybar = 149.75
z = (ybar - mu0)/(sigma/sqrt(n))
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))