Question

A major retailer wants to track the number of products that are returned to the store after purchase. During a particular week the retailer sold a total of 165 products, and 15 items were returned.

(a) Determine a 95% confidence interval for the proportion of returned products sold by the retailer. (Round your answers to four decimal places.)

(b) If 12 thousand products are sold by the retailer in a given year, determine the 95% confidence interval for the number of products (how many) that would be returned. (Round your answers to the nearest whole number.)

Answer 1

To determine a confidence interval for the proportion of returned products sold by the retailer, we calculate the sample proportion and the margin of error. The 95% confidence interval for the proportion of returned products is approximately 0.0512 to 0.1306. If 12,000 products are sold in a year, the 95% confidence interval for the number of products that would be returned is approximately 1043 to 1137.

Step-by-step explanation:

To determine a confidence interval for the proportion of returned products sold by the retailer, we need to calculate the sample proportion and the margin of error.

(a) The sample proportion, or the proportion of returned products, is calculated by dividing the number of returned items by the total number of products sold: 15/165 = 0.0909.

Next, we calculate the margin of error, which is determined using the formula: z * sqrt((p * (1-p))/n), where z is the z-score for the desired confidence level (95% corresponds to a z-score of approximately 1.96), p is the sample proportion, and n is the sample size. Plugging in the values, we get: 1.96 * sqrt((0.0909 * (1-0.0909))/165) = 0.0397.

The confidence interval is then calculated by subtracting and adding the margin of error to the sample proportion: 0.0909 - 0.0397 to 0.0909 + 0.0397. Round the answers to four decimal places to get the confidence interval: (0.0512, 0.1306).

(b) To determine the confidence interval for the number of products that would be returned if 12,000 products are sold in a year, we multiply the sample proportion by the total number of products sold: 0.0909 * 12,000 = 1090.8.

The margin of error is calculated using the formula: z * sqrt((p * (1-p))/n), where z is the z-score for the desired confidence level, p is the sample proportion, and n is the sample size. Plugging in the values, we get: 1.96 * sqrt((0.0909 * (1-0.0909))/165) = 47.6404.

The confidence interval is then calculated by subtracting and adding the margin of error to the number of returned products: 1090.8 - 47.6404 to 1090.8 + 47.6404. Round the answers to the nearest whole number: (1043, 1137).