A bank wants to attract new customers for its credit card. The bank tries two different approaches in the marketing campaign. The first promises a cash back reward; the second promises low interest rates. A sample of 500 people is called the first brochure; of these, 100 get the credit card. A separate sample of 500 people is called the second brochure; 125 get the credit card. The bank wants to know if the two campaigns are equally attractive to customers. What is a 95% confidence interval for the difference in the two proportions

Step-by-step explanation:

Confidence interval for the difference in the two proportions is written as

Difference in sample proportions ± margin of error

Sample proportion, p = x/n

Where x = number of success

n = number of samples

For the first brochure,

x = 100

n1 = 500

p1 = 100/500 = 0.2

For the second brochure

x = 125

n2 = 500

p2 = 125/500 = 0.25

Margin of error = z√[p1 (1 - p1) / n1 + p2 (1 - p2) / n2]

To determine the z score, we subtract the confidence level from 100% to get α

α = 1 - 0.95 = 0.05

α/2 = 0.05/2 = 0.025

This is the area in each tail. Since we want the area in the middle, it becomes

1 - 0.025 = 0.975

The z score corresponding to the area on the z table is 1.96. Thus, the z score for confidence level of 95% is 1.96

Margin of error = 1.96 * √[0.2 (1 - 0.2) / 500 + 0.25 (1 - 0.25) / 500]

= 1.96 * √0.000695

= 0.052

Confidence interval = (0.2 - 0.25) ± 0.052

= - 0.05 ± 0.052