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Chapter 6: Q. 41 (page 378)

Refer to Exercise 37. The ferry company鈥檚 expenses are $20per trip. Define the random variable Yto be the amount of profit (money collected minus expenses) made by the ferry company on a randomly selected trip. That is, Y=M鈭�20.

(a) How does the mean of Yrelate to the mean of M? Justify your answer. What is the practical importance of 渭Y?

(b) How does the standard deviation of Yrelate to the standard deviation of M? Justify your answer. What is the practical importance of 蟽Y?

Short Answer

Expert verified

(a)渭Y=渭M-20

The expected loss per trip is $0.65.

(b)蟽Y=蟽M=6.45

The expected costs are expected to vary by about $6.45from the expected loss pf- $0.65per trip.

Step by step solution

01

Part (a) Step 1: Given Information 

Given:

Y=M-20

02

Part (a) Step 2: Explanation 

Result of previous exercise:

渭M=19.35蟽M=6.45

Property mean:

渭aX+b=a渭X+b

Then we can determine the mean for Yif we know the mean of M:

渭Y=渭M-20=渭M-20=19.35-20=-0.65.

03

Part (a) Step 1: Given Information 

Given:

Y=M-20

04

Part (b) Step 2: Explanation 

Property standard deviation:

蟽aX+b=a蟽X

Result of the previous exercise:

渭M=19.35蟽M=6.45

Then we can determine the standard deviation for Y:

蟽Y=蟽M-20=蟽M=6.45

The practical importance is that the fixed costs do not influence the spread of the distribution of money (profit/money collected).

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Most popular questions from this chapter

Statistics for investing (3.2) Joe's retirement plan invests in stocks through an "index fund" that follows the behavior of the stock market as a whole, as measured by the Standard & Poor's (S&P) 500 stock index. Joe wants to buy a mutual fund that does not track the index closely. He reads that monthly returns from Fidelity Technology Fund have correlation r=0.77with the S&P500index and that Fidelity Real Estate Fund has correlation role="math" localid="1649424494964" r=0.37with the index.

(a) Which of these funds has the closer relationship to returns from the stock market as a whole? How do you know?

(b) Does the information given tell Joe anything about which fund has had higher returns?

7. Benford鈥檚 law Refer to Exercise 5. The first digit of a randomly chosen expense account claim follows Benford鈥檚 law. Consider the events A = first digit is 7 or greater and B = first digit is odd.

(a) What outcomes make up the event A? What is P(A)?

(b) What outcomes make up the event B? What is P(B)?

(c) What outcomes make up the event 鈥淎 or B鈥�? What is P(A or B)? Why is this probability not equal to P(A) + P(B)?

A large auto dealership keeps track of sales made during each hour of the day. Let X= the number of cars sold during the 铿乺st hour of business on a randomly selected Friday. Based on previous records, the probability distribution of Xis as follows:

The random variable Xhas mean 渭X=1.1and standard deviation 蟽X=0.943.

To encourage customers to buy cars on Friday mornings, the manager spends 75to provide coffee and doughnuts. The manager鈥檚 net pro铿乼 Ton a randomly selected Friday is the bonus earned minus this 75. Find the mean and standard deviation ofT.

A small ferry runs every half hour from one side of a large river to the other. The number of cars Xon a randomly chosen ferry trip has the probability distribution shown below. You can check that 渭X=3.87and 蟽X=1.29.

(a) The cost for the ferry trip is $5. Make a graph of the probability distribution for the random variable M=money collected on a randomly selected ferry trip. Describe its shape.

(b) Find and interpret 渭M.

(c) Compute and interpret 蟽M.

Benford鈥檚 law and fraud Refer to Exercise 13. It might also be possible to detect an employee鈥檚 fake expense records by looking at the variability in the first digits of those expense amounts.

(a) Calculate the standard deviation 蟽Y. This gives us an idea of how much variation we鈥檇 expect in the employee鈥檚 expense records if he assumed that first digits from 1 to 9 were equally likely.

(b) Now calculate the standard deviation of first digits that follow Benford鈥檚 law (Exercise 5). Would using standard deviations be a good way to detect fraud? Explain.
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