Question

You may need to use the appropriate appendix table or technology to answer this 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 190 products, and 16 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.) ▭to ▭
(b) If 14 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.) ▭ to▭ products returned

Answer 1

To determine a 95% confidence interval for the proportion of returned products sold by the retailer, we can use the plus-four method. For part (b), if 14,000 products are sold by the retailer in a year, we can use the same formula to calculate the confidence interval for the number of products that would be returned.

Step-by-step explanation:

To determine a 95% confidence interval for the proportion of returned products sold by the retailer, we can use the plus-four method. First, calculate the sample proportion by dividing the number of items returned by the total number of products sold: 16/190 = 0.0842. Next, add four to the number of items returned and add four to the total number of products sold: (16+4)/(190+4) = 0.0903. This gives us the adjusted proportion. Finally, to calculate the confidence interval, we can use the formula:

Adjusted Proportion ± Margin of Error

The margin of error can be calculated as:

z * √((Adjusted Proportion * (1 - Adjusted Proportion)) / n)

where z is the z-score for the desired confidence level (in this case, 1.96 for a 95% confidence level) and n is the total number of products sold (190 in this case). Substituting the values into the formula, we get:

0.0903 ± 1.96 * √((0.0903 * (1 - 0.0903)) / 190)

Calculating this expression gives us the 95% confidence interval for the proportion of returned products sold by the retailer.

For part (b) of the question, if 14,000 products are sold by the retailer in a year, we can use the same formula to calculate the confidence interval for the number of products that would be returned. However, instead of using the sample proportion, we would use the estimated proportion, which can be calculated as the number of items returned divided by the total number of products sold in a year. The estimated proportion would then be used in the formula to calculate the confidence interval.