Question

The following questions ask you to compute a point estimate and confidence interval for a difference in proportions, p1 - p2. We let n1 denote the number of trials and x1 the number of observed successes in the first sample; and n2 denote the number of trials and x2 the number of observed successes in the second sample.

a. n1 = 10, x1 = 7, n2 = 50, x2 = 5. Compute a point estimate and 95% confidence interval for p1 - p2.

b. n1 = 50, x1 = 24, n2 = 55, x2 = 27 . Compute a point estimate and 95% confidence interval for p1-p2.

Answer 1

a)
Here, , n1 = 10 , n2 = 50
p1cap = 0.7 , p2cap = 0.1

point estimate = p1cap - p2cap = 0.7 - 0.1 = 0.6

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.7 * (1-0.7)/10 + 0.1*(1-0.1)/50)
SE = 0.151

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.7 - 0.1 - 1.96*0.151, 0.7 - 0.1 + 1.96*0.151)
CI = (0.304 , 0.896)

b)

Here, , n1 = 50 , n2 = 55
p1cap = 0.48 , p2cap = 0.4909

point estimate = p1cap - p2cap
= 0.48 - 0.4909
= -0.0109

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.48 * (1-0.48)/50 + 0.4909*(1-0.4909)/55)
SE = 0.0977

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.48 - 0.4909 - 1.96*0.0977, 0.48 - 0.4909 + 1.96*0.0977)
CI = (-0.2024 , 0.1806)