Statistics Question

Question

The average demand for rental skis on winter Saturdays at a particular area is 150 pairs which has been fairly stable over time. There is variation due to weather conditions and competing areas; the standard deviation is 20 pairs. The demand distribution seems to be roughly normal.

A.) The rental shop stocks 170 pairs of skis. What is the probability that demand will exceed this supply on any winter Saturday?

B.) How many pairs of skis in stock does the shop have to have to make the probability in question A less than .01?

I understand question A, but I do not know how to get to the answer to B. I know that the z-score for B is 1.28 and that the answer is 175.6, but I do not understand the steps to get to the answer.

Answer 1

Good question. Z-scores are measured in standard deviations (σ). So a z-score of +2 would be 2 σ above the mean, and a z-score of -1.5 would be 1.5 σ below the mean (below because it's negative). Your situation with a z-score of +1.28 means it falls 1.28 σ above the mean. Since σ is 20 pairs of skis, that's 1.28 * 20 = 25.6 pairs of skis above the mean. The mean is 150 pairs of skis, so 1.28 σ above that is 175.6 pairs of skis.

"I know that the z-score for B is 1.28 and that the answer is 175.6, but I do not understand the steps to get to the answer."