Question

A small accounting firm recently leased a new document management device (an all-in-one machine capable of printing, scanning, copying, and faxing) from a supplier for three years. Reliability was a deciding factor. The company’s research found that during any given year, approximately 86% of the devices sold in one particular model required no service in a year, 9% needed one repair, 4% needed two repairs, 1% required three repairs, and none required more than three repairs in a year.

In addition to the lease, the supplier offers a maintenance contract for the device for a fixed cost of $100 per year plus a $25 service charge per repair. Considering the information above to suggest a probability distribution for the number of repairs, use the rules of variance to find the standard deviation of the yearly repair cost.

a. $187.42

b. $105

c. $10.25

d. $2.75

e. 13.69

Please explain the steps.

Answer 1

Solution:

Given: The company’s research found that during any given year, approximately 86% of the devices sold in one particular model required no service in a year, 9% needed one repair, 4% needed two repairs, 1% required three repairs, and none required more than three repairs in a year.

Thus we have:

x: Number of Repairs	P(x) : Probability
0	0.86
1	0.09
2	0.04
3	0.01
Total   =	1

Also we are given:

The supplier offers a maintenance contract for the device for a fixed cost of $100 per year plus a $25 service charge per repair.

We have to find the standard deviation of the yearly repair cost.

Let y = repair cost = 100 + 25 * x

First find Variance of x.

Thus we need to make following table:

x: Number of Repairs	P(x) : Probability	x*P(x)	x^2*P(x)
0	0.86	0	0
1	0.09	0.09	0.09
2	0.04	0.08	0.16
3	0.01	0.03	0.09
Total   =	1

Thus

and

Thus

Now find variance of Y = Repair cost

Use following rules of Variance:

Variance of constant is 0, that is if A is constant , then Var(A) =0

and Var(AX) = A² * Var(X)

Now find standard deviation of Y:

Thus correct option is: e. 13.69