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 175 products, and 17 items were returned

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

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.)

Answer 1

The 95% confidence interval for the proportion of returned products is [0.0520, 0.1422]. For a yearly projection of 14,000 products sold, the confidence interval for the number of returned products is [728, 1989], when rounded to the nearest whole number.

Step-by-step explanation:

To find the 95% confidence interval for the proportion of returned products sold by the retailer, we can use the formula for a confidence interval for a population proportion:

CI = π ± Z*sqrt(π(1-π)/n)

where:
π = sample proportion of returns = 17/175
Z* = Z-value for 95% confidence interval (approximately 1.96)
n = sample size = 175

Plugging in our values:

π = 17/175 = 0.0971
CI = 0.0971 ± 1.96*sqrt(0.0971(1-0.0971)/175)
CI = 0.0971 ± 0.0451
CI = [0.0520, 0.1422]

Rounded to four decimal places, the confidence interval is [0.0520, 0.1422].

For the yearly projection, if 14,000 products are sold, the interval in terms of the number of products (rather than proportion) can be calculated by multiplying each end of the proportion interval by 14,000:

Lower bound = 0.0520 * 14,000 ≈ 728
Upper bound = 0.1422 * 14,000 ≈ 1989

Rounded to the nearest whole number, the confidence interval for the number of products that would be returned in a year is [728, 1989].