Question

Construct the indicated confidence interval for the difference between population proportions p₁-p₂ . Assume that the samples are independent and that they have been randomly selected.

x₁ = 11, n₁ = 45 and x₂ = 18, n₂ = 54; Construct a 90% confidence interval for the difference between population proportions p₁ - p₂.

0.094 <p₁ - p₂ < 0.395

0.067 <p₁ - p₂ < 0.422

-0.238 <p₁ - p₂ < 0.060

0.422 <p₁ - p₂ < 0.065

Answer 1

For Sample 1:

n1 = 45, x1 = 11

p̂1 = x1/n1 = 0.2444

For Sample 2:

n2 = 54, x2 = 18

p̂2 = x2/n2 = 0.3333

90% Confidence interval for the difference:

At α = 0.1, two tailed critical value, z_c = NORM.S.INV(0.1/2) = 1.645

Lower Bound = (p̂1 - p̂2) - z_c*√ [(p̂1*(1-p̂1)/n1)+(p̂2*(1-p̂2)/n2) ] = (0.2444 - 0.3333) - 1.645*√[(0.2444*0.7556/45) + (0.3333*0.6667/54)] = -0.238

Upper Bound = (p̂1 - p̂2) + z_c*√ [(p̂1*(1-p̂1)/n1)+(p̂2*(1-p̂2)/n2) ] = (0.2444 - 0.3333) + 1.645*√[(0.2444*0.7556/45) + (0.3333*0.6667/54)] = 0.060

-0.238 < p1 -p2 < 0.060