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Chapter 4: Q73E (page 247)

Contaminated gun cartridges. Refer to the investigation of contaminated gun cartridges at a weapons manufacturer, presented in Exercise 4.29 (p. 226). In a sample of 158 cartridges from a certain lot, 36 were found to be contaminated, and 122 were 鈥渃lean.鈥� If you randomly select 5 of these 158 cartridges, what is the probability that all 5 will be 鈥渃lean鈥�?

Short Answer

Expert verified

The probability that all 5 will be clean is 0.20.

Step by step solution

01

Given Information

A lot contains 158 cartridges.

N=158M=36N-M=122n=5

02

Hypergeometric Distribution is used to find the probability

The probability is calculated as

px=0=M0脳N-MN-0Nn=360脳1225-01580=1脳207288004769754986=0.26

Therefore, the probability is 0.26.

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Most popular questions from this chapter

Errors in measuring truck weights. To help highway planners anticipate the need for road repairs and design futureconstruction projects, data are collected on the estimatedvolume and weight of truck traffic on specific roadways (Transportation Planning Handbook, 2016) using specialized 鈥渨eigh-in-motion鈥� equipment. In an experiment involving repeated weighing of a 27,907-pound truck, it wasfound that the weights recorded by the weigh-in-motionequipment were approximately normally distributed witha mean of 27,315 and a standard deviation of 628 pounds

(Minnesota Department of Transportation). It follows thatthe difference between the actual weight and recordedweight, the error of measurement, is normally distributedwith a mean of 592 pounds and a standard deviation of 628pounds.

a. What is the probability that the weigh-in-motion equipment understates the actual weight of the truck?

b. If a 27,907-pound truck was driven over the weigh-in-motion equipment 100 times, approximately howmany times would the equipment overstate the truck鈥檚weight?

c. What is the probability that the error in the weightrecorded by the weigh-in-motion equipment for a27,907-pound truck exceeds 400 pounds?

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The Apprenticecontestants鈥� performance ratings. Referto the Significance(April 2015) study of contestants鈥� performanceson the United Kingdom鈥檚 version of the TVshow, The Apprentice, Exercise 2.9 (p. 73). Recall thatthe performance of each of 159 contestants was rated ona 20-point scale. Contestants were also divided into twogroups: those who played for a job and those who playedfor a businesspartnership. These data (simulated, based onstatistics reportedin the article) are saved in the accompanyingfile. Descriptive statistics for each of the two groupsof contestants are displayed in the accompanying Minitabprintout.

a. Determine whether the performance ratings of contestantswho played for a job are approximately normallydistributed.

b. Determine whether the performance ratings of contestantswho played for a business partnership are approximatelynormally distributed.

Descriptive Statistics: Rating

Variable Rating

Price

N

Mean

St.Dev

Minimum

Q

1

median

Q3

Maximum

IQ

R

Job

99

7.879

4.224

1

4

9

11

20

7

Partner

60

8.883

4.809

1

5

8

12

20

7

Buy-side vs. sell-side analysts鈥� earnings forecasts. Financial analysts who make forecasts of stock prices are categorized as either 鈥渂uy-side鈥� analysts or 鈥渟ell-side鈥� analysts. Refer to the Financial Analysts Journal (July/August 2008) comparison of earnings forecasts of buy-side and sell-side analysts, Exercise 2.86 (p. 112). The mean and standard deviation of forecast errors for both types of analysts are reproduced in the table. Assume that the distribution of forecast errors are approximately normally distributed.

a. Find the probability that a buy-side analyst has a forecast error of +2.00 or higher.

b. Find the probability that a sell-side analyst has a forecast error of +2.00 or higher


Buy-Side Analysts

Sell-Side Analysts

Mean

0.85

-0.05

Standard Deviation

1.93

0.85

Purchasing decision. Suppose you are a purchasing officer for a large company. You have purchased 5 million electrical switches, and your supplier has guaranteed that the shipment will contain no more than .1% defectives. To check the shipment, you randomly sample 500 switches, test them, and find that four are defective. Based on this evidence, do you think the supplier has complied with the guarantee? Explain

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a. Demonstrate that x is (approximately) a binomial randomvariable.

b. Give the values of p and n for the binomial distribution.

c. Find P(x=15), the probability that SBIRS will detect the object on exactly 15 tracks.

d. Find P(x鈮�15), the probability that SBIRS will detect the object on at least 15 tracks.

e. FindE(x) and interpret the result.
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