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Chapter 6: Q. 100 (page 406)

Using Benford's law According to Benford's law (Exercise 5, page 353), the probability that the first digit of the amount of a randomly chosen invoice is an 8or a 9is 0.097. Suppose you examine randomly selected invoices from a vendor until you find one whose amount begins with an 8or a 9.

(a) How many invoices do you expect to examine until you get one that begins with an 8or 9? Justify your answer.

(b) In fact, you don't get an amount starting with an 8or 9until the 40thinvoice. Do you suspect that the invoice amounts are not genuine? Compute an appropriate probability to support your answer.

Short Answer

Expert verified

a. The number of invoices expected to examine until begins with an 8or 9is 10.3093invoices.

b. The invoice amounts are not genuine.

Step by step solution

01

Part (a) Step 1: Given Information 

The probability=0.097

Randomly selected invoices from a vendor until you find one whose amount begins=8or9.

02

Part (a) Step 2: Explanation 

Given:

p=0.097

Probability (or mean) of a geometric variable is the reciprocal of the expected number:

渭=1p

=10.097

鈮�10.3093

03

Part (b) Step 1: Given Information

The probability=0.097

Randomly selected invoices from a vendor until you find one whose amount begins=8or9.

04

Part (b) Step 2: Explanation 

Given:

p=0.097

Geometric probability formula:

P(X=k)=qk鈭�1p=(1鈭�p)k鈭�1p

Find the probability using the complement rule:

P(X鈮�40)=1鈭�P(X<40)=1鈭�P(X=1)鈭�P(X=2)鈭�鈥�鈭�P(X=39)=1鈭�0.9813鈮�0.0187=1.87%

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Most popular questions from this chapter

Exercises 47 and 48 refer to the following setting. Two independent random variables Xand Yhave the probability distributions, means, and standard deviations shown.

48. Difference Let the random variable D=X-Y.

(a) Find all possible values of D. Compute the probability that Dtakes each of these values. Summarize the probability distribution ofD in a table.
(b) Show that the mean of Dis equal to渭X-渭Y.
(c) Confirm that the variance of Dis equal to 蟽X2+蟽Y2.Find all possible values of D. Compute the probability that takes each of these values. Summarize the probability distribution of in a table.

In which of the following situations would it be appropriate to use a Normal distribution to approximate probabilities for a binomial distribution with the given values of n and p?

(a)n=10,p=0.5

(b)n=40,p=0.88

(c)n=100,p=0.2

(d)n=100,p=0.99

(e)n=1000,p=0.003

What鈥檚 the expected number of cars in a randomly selected American household?

(a) Between 0and 5

(b) 1.00

(c) 1.75

(d) 1.84

(e) 2.00

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?

Refer to the previous Check Your Understanding (page 390) about Mrs. Desai's special multiple-choice quiz on binomial distributions. We defined X=the number of Patti's correct guesses.

3. What's the probability that the number of Patti's correct guesses is more than 2standard deviations above the mean? Show your method.
See all solutions

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