two_sample_mean_difference_ci = function(x_data, y_data, alpha, equal_var=TRUE){
    n = length(x_data)
    xbar = mean(x_data)
    sx2 = var(x_data)

    m = length(y_data)
    ybar = mean(y_data)
    sy2 = var(y_data)

    if( equal_var ){
        sp2 = ( (n-1)*sx2 + (m-1)*sy2 ) / ( n+m-2 )
        nu = n+m-2
        se = sqrt(sp2 * (1/n + 1/m))
    }else{
        theta_hat = sx2/sy2
        numer = (theta_hat + n/m)^2
        denom = theta_hat^2/(n-1) + (n/m)^2/(m-1)
        nu = round(numer/denom)
        se = sqrt( sx2/n + sy2/m )
    }

    t_crit = qt( 1-alpha/2, nu )
    ci = xbar - ybar + t_crit * se * c(-1, +1)
}


two_sample_variance_ratio_ci = function(x_data, y_data, alpha){
    n = length(x_data)
    sx2 = var(x_data)

    m = length(y_data)
    sy2 = var(y_data)

    f = sx2/sy2

    f_crit_L = qf( alpha/2, m-1, n-1 )
    f_crit_R = qf( 1-alpha/2, m-1, n-1 )

    ci = f * c( f_crit_L, f_crit_R )
}


success_probability_difference_ci = function(x, n, y, m, alpha){
    p_x = x/n
    p_y = y/m

    z_crit = qnorm(1-alpha/2)

    var_x = p_x*(1- p_x)
    var_y = p_y*(1- p_y)
    se = sqrt( var_x/n + var_y/m )
    ci = p_x - p_y + z_crit * se * c(-1, +1)
}



# Section 5; Question 1:
#
early_coinage = c( 5.9, 6.8, 6.4, 7.0, 6.6, 7.7, 7.2, 6.9, 6.2 )
late_coinage = c( 5.3, 5.6, 5.5, 5.1, 6.2, 5.8, 5.8 )

ci = two_sample_mean_difference_ci(early_coinage, late_coinage, alpha = 0.1)
print(ci)


# Section 5; Question 2:
#
with_dome = c( 100.0, 58.6, 93.5, 83.6, 84.1 )
without_dome = c( 76.4, 84.2, 96.5, 88.8, 85.3, 79.1, 83.6 )

ci = two_sample_mean_difference_ci(early_coinage, late_coinage, alpha = 0.05)
print(ci)


# Section 5; Question 3:
#
source('case_study_9_2_3_data.R')
ci = two_sample_mean_difference_ci(large_sales, small_sales, alpha = 0.01)

print(sprintf('Equal variances: ci= (%f, %f)', ci[1], ci[2]))
ci = two_sample_mean_difference_ci(large_sales, small_sales, alpha = 0.01, equal_var=FALSE)
print(sprintf('Unequal variances: ci= (%f, %f)', ci[1], ci[2]))


# Section 5; Question 6:
#
source('case_study_9_2_1_data.R' )

ci = two_sample_variance_ratio_ci(DF[DF$author=='Twain',]$prop, DF[DF$author=='QCS', ]$prop, alpha= 0.05)
print(ci)


# Section 5; Question 7:
#
normal = c( 62, 60, 78, 62, 49, 67, 80, 48 )
pericarditis = c( 24, 56, 42, 74, 44, 28 )

ci = two_sample_variance_ratio_ci(normal, pericarditis, alpha=0.05)
print(ci)


# Section 5; Question 9:
#
ci = success_probability_difference_ci(126, 782, 111, 758, alpha=0.05)
print(ci)


# Section 5; Question 10:
#
x = 55 # men
n = 160
y = 60 # women
m = 192

ci = success_probability_difference_ci(x, n, y, m, alpha=0.2)
print(ci)


# Section 5; Question 12:
#
ci = success_probability_difference_ci(106, 3522, 13, 115, alpha=0.05)
print(ci)