Exam 14: Random Variables

A fast food restaurant just leased a new freezer and food fryer for three years. The service contract for the freezer offers unlimited repairs for a fee of $125 a year plus a $35 service charge for each repair needed. The restaurant’s research suggested that during a given year 80% of these freezers need no repairs, 11% needed to be serviced once, 5% twice, 4% three times, and none required more than three repairs. -Which service contract should the restaurant expect to cost more each year? How much more? With what standard deviation?

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Question 1

Correct Answer:

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The food fryer's service contract is expected to cost more.
$E ($ fryer $-$ freezer $) = \$ 140 - \$ 136.55 = \$ 3.45$ more
$\operatorname { Var } ($ fryer - freezer $) = \$ 47.83$ (same as the sum in problem 6 )

A fast food restaurant just leased a new freezer and food fryer for three years. The service contract for the freezer offers unlimited repairs for a fee of $125 a year plus a $35 service charge for each repair needed. The restaurant’s research suggested that during a given year 80% of these freezers need no repairs, 11% needed to be serviced once, 5% twice, 4% three times, and none required more than three repairs. -What is the standard deviation of the number of repairs that may be required during the three- year term of the lease? On what assumption does your calculation rest? Do you think this assumption is reasonable?

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Question 2

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$\operatorname { Var } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right) = 0.561 + 0.561 + 0.561 = 1.683$ , so standard deviation $( C ) = 1.297$
The assumption is that the number of repairs is independent from year to year. This might be incorrect because some freezers might need more service than others.

A fast food restaurant just leased a new freezer and food fryer for three years. The service contract for the freezer offers unlimited repairs for a fee of $125 a year plus a $35 service charge for each repair needed. The restaurant’s research suggested that during a given year 80% of these freezers need no repairs, 11% needed to be serviced once, 5% twice, 4% three times, and none required more than three repairs. -How many times should the restaurant expect to have to get this freezer repaired over the three-year term of the lease?

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Question 3

Correct Answer:

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$E \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right) = 0.33 + 0.33 + 0.33 = 0.99 \text { repairs }$

A fast food restaurant just leased a new freezer and food fryer for three years. The service contract for the freezer offers unlimited repairs for a fee of $125 a year plus a $35 service charge for each repair needed. The restaurant’s research suggested that during a given year 80% of these freezers need no repairs, 11% needed to be serviced once, 5% twice, 4% three times, and none required more than three repairs. -Find the standard deviation of the number of repairs each year.

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Question 4

A fast food restaurant just leased a new freezer and food fryer for three years. The service contract for the freezer offers unlimited repairs for a fee of $125 a year plus a $35 service charge for each repair needed. The restaurant’s research suggested that during a given year 80% of these freezers need no repairs, 11% needed to be serviced once, 5% twice, 4% three times, and none required more than three repairs. -What are the mean and standard deviation of the restaurant's annual expense for the service contract?

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Question 5

A fast food restaurant just leased a new freezer and food fryer for three years. The service contract for the freezer offers unlimited repairs for a fee of $125 a year plus a $35 service charge for each repair needed. The restaurant’s research suggested that during a given year 80% of these freezers need no repairs, 11% needed to be serviced once, 5% twice, 4% three times, and none required more than three repairs. -The yearly service contract for the food fryer estimates a mean annual cost of $140 with a standard deviation of $40.What is the expected value and standard deviation of the total cost for the service contracts for the freezer and the food fryer?

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