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Chapter 4: Q107E (page 263)

Mean shifts on a production line. Six Sigma is a comprehensive approach to quality goal setting that involves statistics. An article in Aircraft Engineering and Aerospace Technology (Vol. 76, No. 6, 2004) demonstrated the use of the normal distribution in Six Sigma goal setting at Motorola Corporation. Motorola discovered that the average defect rate for parts produced on an assembly line varies from run to run and is approximately normally distributed with a mean equal to 3 defects per million. Assume that the goal at Motorola is for the average defect rate to vary no more than 1.5 standard deviations above or below the mean of 3. How likely is it that the goal will be met?

Short Answer

Expert verified

There is an 87% chance that the goal will be met

Step by step solution

01

Given information

Assume that Motorola discovered the average defect rate for parts produced on an assembly line varies from run to run and it is approximately normally distributed with a mean of 3 and standard deviation sigma.

02

Calculating for probability

Here, x follows a normal distribution with μ=3andσ=σ

x=3+1.5σ

The z-score is,

z=x-μσ=3+1.5σ-3σ=1.5σσ=1.5

Again,x=3-1.5σ

z=x-μσ=3-1.5σ-3σ=-1.5σσ=-1.5

The probability is,

P(μ-1.5σ<x<μ+1.5σ)=P(3-1.5σ<x<3+1.5σ)=P(x<3+1.5σ)-P(x<31.5σ)=P(z<1.5)-(1-P(z<1.5))=0.9332-1+0.9332=0.8664≈0.87P(μ-1.5σ<x<μ+1.5σ)=0.87a

Therefore, there is an 87% chance that the goal will be met

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Most popular questions from this chapter

178 Dutch elm disease. A nursery advertises that it has 10 elm treesfor sale. Unknown to the nursery, 3 of the trees have already been infected with Dutch elm disease and will die withina year.

a. If a buyer purchases 2 trees, what is the probability that bothtrees will be healthy?

b. Refer to part a. What is the probability that at least 1 of thetrees is infected?

4.113 Credit/debit card market shares. The following table reports the U.S. credit/debit card industry’s market share data for 2015. A random sample of 100 credit/debit card users is to be questioned regarding their satisfaction with

their card company. For simplification, assume that each card user carries just one card and that the market share percentages are the percentages of all card customers that carry each brand.

Credit debit Card

Market Share %

Visa

59

MasterCard

26

Discover

2

American Express

13

Source:Based on Nilson Reportdata, June 2015.

a. Propose a procedure for randomly selecting the 100 card users.

b. For random samples of 100 card users, what is the expected number of customers who carry Visa? Discover?

c. What is the approximate probability that half or more of the sample of card users carry Visa? American Express?

d. Justify the use of the normal approximation to the binomial in answering the question in part c.

Traffic fatalities and sporting events. The relationship betweenclose sporting events and game-day traffic fatalities was investigated in the Journal of Consumer Research (December 2011). The researchers found that closer football and basketball games are associated with more traffic fatalities. The methodology used by the researchers involvedmodeling the traffic fatality count for a particular game as a Poisson random variable. For games played at the winner’s location (home court or home field), the mean number of traffic fatalities was .5. Use this information to find the probability that at least three game-day traffic fatalities will occur at the winning team’s location.

How many questionnaires to mail? The probability that a consumer responds to a marketing department’s mailed questionnaire is 0.4. How many questionnaires should be mailed if you want to be reasonably certain that at least 100 will be returned?

Public transit deaths. Millions of suburban commuters use the public transit system (e.g., subway trains) as an alter native to the automobile. While generally perceived as a safe mode of transportation, the average number of deaths per week due to public transit accidents is 5 (Bureau of Transportation Statistics, 2015).

a. Construct arguments both for and against the use of the Poisson distribution to characterize the number of deaths per week due to public transit accidents.

b. For the remainder of this exercise, assume the Poisson distribution is an adequate approximation for x, the number of deaths per week due to public transit accidents. Find E(x)and the standard deviation of x.

c. Based strictly on your answers to part b, is it likely that more than 12 deaths occur next week? Explain.

d. Findp(x>12). Is this probability consistent with your answer to part c? Explain.
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