Question

At a certain coffee​ shop, all the customers buy a cup of coffee and some also buy a doughnut. The shop owner believes that the number of cups he sells each day is normally distributed with a mean of 310 cups and a standard deviation of 22 cups. He also believes that the number of doughnuts he sells each day is independent of the coffee sales and is normally distributed with a mean of 180 doughnuts and a standard deviation of 15. b) If he makes a profit of 50 cents on each cup of coffee and 40 cents on each​ doughnut, can he reasonably expect to have a​ day's profit of over​ $300? Explain.

A. No.​ $300 is more than 5 standard deviations above the mean.

B. No. The number of doughnuts he expects to sell plus the number of cups of coffee is less than 600.

C. Yes.​ $300 is less than 6 standard deviations above the mean.

D. Yes. The number of doughnuts he expects to sell plus the number of cups of coffee is greater than 300.

Answer 1

Please don't hesitate to give a "thumbs up" in case you're satisfied with the answer

So, everything beyond 2 deviation is unusual.

Lets see how much is the expected value at 2 deviation above mean

For coffee, expected mean of the combined distribution = Mean 1 + Mean 2 = .5*310 + .4*180 = $227

For donut, expected mean of the combined distribution = sqrt((.5*22)^2 + (.4*15)^2) = $12.53

So, $300 is (300-227)/12.53 = 5.83 deviations away from mean

So, A is correct. It is unusual to expect $300