Question

Equifax, a credit rating agency, wants an estimate of the percentage of delinquent credit cards (where the customer has not made at least the minimum monthly payment by the due date). The manager randomly selects 150 credit card accounts and finds 9 are delinquent.

(a) Give a point estimate of the percentage of delinquent credit cards.
(b) Construct a 95% confidence interval estimate for the percentage of delinquent credit cards.
(c) What is the margin of error on the estimate?
(d) What is the worst-case estimate for the percentage delinquent credit cards?
(e) The manager wants to reduce the error on the estimate of the percentage of delinquent credit cards to 2% with 95% confidence. How many
accounts need to be sampled?
(f) Equifax also wants an estimate of the percentage of delinquent credit cards with a balance owing of at least $5,000. They want the error on the estimate to be no more than 2% with 95% confidence. How many delinquent accounts would need to be sampled?

Answer 1

(a) The point estimate of the percentage of delinquent credit cards is 6%.

(b) The 95% confidence interval estimate for the percentage of delinquent credit cards is approximately 2.89% to 9.11%.

Thus the correct option is a and b.

Step-by-step explanation:

(a) The point estimate is obtained by taking the ratio of delinquent credit cards to the total number of sampled accounts:
`\( (9)/(150) = 0.06 \),` which is equivalent to 6%.

(b) To construct a 95% confidence interval, the margin of error is calculated based on the standard error of the proportion.

Using the formula
`\( \text{Margin of Error} = Z \`times
`\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \),`where
`\( Z \)` is the z-score for a 95% confidence level,
`\( \hat{p} \)` is the point estimate, and
`\( n \)` is the sample size, we can find the margin of error.

Adding and subtracting this margin of error from the point estimate gives the confidence interval.

This interval provides a range within which we can be 95% confident that the true percentage of delinquent credit cards lies.

(c) The margin of error is the range above and below the point estimate within which the true population proportion is likely to fall.

(d) The worst-case estimate is the maximum percentage of delinquent credit cards, which occurs when all 150 sampled accounts are delinquent, resulting in a worst-case estimate of 100%.

(e) To achieve a margin of error of 2%, the necessary sample size can be calculated using the margin of error formula, rearranged to solve for ( n ). After finding ( n ), the manager would need to sample that many accounts to achieve the desired precision in the estimate.

(f) The process for determining the sample size in (e) can be applied to estimate the number of delinquent accounts with a balance over $5,000 while maintaining a 2% margin of error at a 95% confidence level.