Quality Progress, February 2005, reports on the results achieved by Bank of American in improving customer satisfaction and customer loyalty by listening to the "voice of the customer". A key measure of customer satisfaction is the response on a scale from 1 to 10 to the question: "Considering all the business you do with Bank of America, what is your overall satisfaction with Bank of America?" Suppose that a random sample of 350 current customers results in 195 customers with a response of 9 or 10 representing "customer delight". Find a 95 percent confidence interval for the true proportion of all current Bank of America customers who would respond with a 9 or 10. Are we 95 percent confident that this proportion exceeds .48, the historical proportion of customer delight for Bank of America?

Question

Quality Progress, February 2005, reports on the results achieved by Bank of American in improving customer satisfaction and customer loyalty by listening to the "voice of the customer". A key measure of customer satisfaction is the response on a scale from 1 to 10 to the question: "Considering all the business you do with Bank of America, what is your overall satisfaction with Bank of America?" Suppose that a random sample of 350 current customers results in 195 customers with a response of 9 or 10 representing "customer delight". Find a 95 percent confidence interval for the true proportion of all current Bank of America customers who would respond with a 9 or 10. Are we 95 percent confident that this proportion exceeds .48, the historical proportion of customer delight for Bank of America?

Answer 1

To find a 95 percent confidence interval for the true proportion of all current Bank of America customers who would respond with a 9 or 10, we can use the formula for confidence interval for a proportion:

Confidence Interval = Sample Proportion ± Margin of Error

The sample proportion is calculated by dividing the number of customers who responded with a 9 or 10 (195) by the total sample size (350):

Sample Proportion = 195 / 350 = 0.5571

To determine the margin of error, we need to use the critical value for a 95 percent confidence level. Since we are dealing with proportions, we can use the Z-distribution and the formula for margin of error is:

Margin of Error = Z * sqrt((Sample Proportion * (1 - Sample Proportion)) / Sample Size)

The critical value for a 95 percent confidence level is 1.96 (which can be obtained from a Z-table or calculator).

Sample Size = 350

Plugging in the values:

Margin of Error = 1.96 * sqrt((0.5571 * (1 - 0.5571)) / 350)

Calculating the margin of error:

Margin of Error = 0.0466

We can now calculate the confidence interval:

Confidence Interval = 0.5571 ± 0.0466

Confidence Interval = (0.5105, 0.6037)

The 95 percent confidence interval for the true proportion of all current Bank of America customers who would respond with a 9 or 10 is (0.5105, 0.6037).

To determine whether we are 95 percent confident that this proportion exceeds 0.48, the historical proportion of customer delight for Bank of America, we can check if this proportion falls outside the confidence interval.

Since the lower value of the confidence interval (0.5105) is greater than 0.48, we can conclude with 95 percent confidence that the true proportion of all current Bank of America customers who would respond with a 9 or 10 exceeds 0.48.