# pset 9
# problem 1 ----
# problem 1b
xbar = 140/250
n = 250
s = 1/2
xbar + qnorm(c(.1,.9))*s/sqrt(n)
qnorm(0.025)
xbar + qnorm(c(.025,.975))*s/sqrt(n)
xbar + c(-2,2)*s/sqrt(n)
xbar + qnorm(c(.1,.9))*s/sqrt(n)

# problem 1c
c(qbeta(.1, 141, 111), qbeta(.9, 141, 111))

# problem 2 ----
# problem 2a
xbar = 69.55
s2 = 14.26
n = 20
df = n-1
alpha = .1
t_alpha2 = qt(1-alpha/2,df)
xbar + c(-t_alpha2, t_alpha2)*sqrt(s2/n)
# Thinking about signficant digits
xbarrange = xbar + c(-.005, .005)
s2max = s2+.005
cilimitsrange = xbarrange + c(-t_alpha2,t_alpha2)*sqrt(s2max/n)
cilimitsrange

# problem 2b
xbar = 69.55
s2 = 3.77^2
n = 20
alpha = .1
z_alpha2 = qnorm(1-alpha/2,0,1)
xbar + c(-z_alpha2, z_alpha2)*sqrt(s2/n)
# Thinking about signficant digits
xbarrange = xbar + c(-.005, .005)
cilimitsrange = xbarrange + c(-z_alpha2,z_alpha2)*sqrt(s2/n)
cilimitsrange

#problem 2c
s2 = 3.77^2
alpha = .1
z_alpha2 = qnorm(1-alpha/2,0,1)
4*s2*z_alpha2^2

#problem 2d
s2 = 14.26
alpha = .1
for (n in 130:180)
{
  t_alpha2 = qt(1-alpha/2,n-1)
  width = 2*t_alpha2*sqrt(s2/n)
  s = sprintf("n = %d, width = %.3f", n, width)
  print(s)
}

# problem 3 ----
# problem 3a
x = c(352,351,361,353,352,358,360,358,359)
n = length(x)
xbar = mean(x)
s = 3
alpha = .05
z_alpha2 = qnorm(1-alpha/2,0,1)
xbar + c(-z_alpha2, z_alpha2)*s/sqrt(n)

# problem 3b
x = c(352,351,361,353,352,358,360,358,359)
n = length(x)
xbar = mean(x)
s = 3
alpha = .02
z_alpha2 = qnorm(1-alpha/2,0,1)
xbar + c(-z_alpha2, z_alpha2)*s/sqrt(n)

# problem 3c
x = c(352,351,361,353,352,358,360,358,359)
n = length(x)
xbar = mean(x)
s = sqrt(var(x))
alpha = .05
t_alpha2 = qt(1-alpha/2, n-1)
xbar + c(-t_alpha2, t_alpha2)*s/sqrt(n)

alpha = .02
t_alpha2 = qt(1-alpha/2, n-1)
xbar + c(-t_alpha2, t_alpha2)*s/sqrt(n)

# problem 4 ----
x = c(6.0, 6.4, 7.0, 5.8, 6.0, 5.8, 5.9, 6.7, 6.1, 6.5, 6.3, 5.8)
n = length(x)
s2 = var(x)
alpha = .05
c_alpha2 = qchisq(1-alpha/2,n-1)
c_oneminusalpha2 = qchisq(alpha/2,n-1)
(n-1)*s2/c(c_alpha2,c_oneminusalpha2)

# problem 5 ----
n = 10
s2 = 4.2
alpha = .05
c_alpha2 = qchisq(1-alpha/2,n-1)
c_oneminusalpha2 = qchisq(alpha/2,n-1)
sqrt((n-1)*s2/c(c_alpha2,c_oneminusalpha2))