A haunted house (not a real one, but one filled with props and scary lights) cos

Question

A haunted house (not a real one, but one filled with props and scary lights) costs $9 in maintenance to run and service every time a customer pays to go through it. The house keeps track of sales from all customers and finds that sale amounts (including admission, treats, and trinkets purchased) are normally distributed with a mean of $13.13, with a standard deviation of $3.13. In order to make a profit, the sales from a customer have to exceed the cost of going in the haunted house.

What is the probability that any one randomly selected customer’s visit to the haunted house will result in a profit?

Frank N. Stein, a college student, works at the haunted house on weekends. In order to participate in profit-sharing, Frank needs the thirteen customers (i.e., parties) who visit on his shift to have combined sales of more than $117.00 ($9.00 per customer). What is the probability that the thirteen parties will, on average, result in a profit for Frank?

Explanation / Answer

= 13.13

= 3.13

a) P(X > 9) = P((X - mean)/sd > (9 - mean)/sd)

= P(Z > (9-13.13)/3.13)

= P(Z > -1.32)

= 1 - P(Z < -1.32)

= 1 - 0.0934

= 0.9066

b) n = 13

117/13 = 9

P(X > 9) = P((X - mean)/(sd/sqrt(n)) > (9 - mean)/(sd/sqrt(n)))

= P(Z > (9 - 13.13)/(3.13/sqrt(13)))

= P(Z > -4.75)

= 1 - P(Z < -4.75)

= 1 - 0

= 1

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