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Chapter 2: Q94E (page 116)

Suppose that 40 and 90 are two elements of a population data set and that their z-scores are -2 and 3, respectively. Using only this information, is it possible to determine the population’s mean and standard deviation? If so, find them. If not, explain why it’s not possible.

Short Answer

Expert verified

Yes, it is possible to find the mean and the standard deviation.

The mean is 60.

The standard deviation is 10.

Step by step solution

01

Given information

x = 40, and x = 90.

The z-score for 40 is (-2), and the z-score for 90 is 3

02

Finding the mean and the standard deviation

As we have two values from the population, it is possible to calculate the mean and the standard deviation.

z-score=x-μσWhenZ=-2,-2=40-μσWhenZ=3,3=90-μσSowecanalsowritetwoequationsmentionedaboveasfollows:-2σ=40-μ3σ=90-μNow,bysubtractingtheequations,weget-2σ-3σ=40-μ-90-μ-5σ=-50σ=505σ=10

Therefore, the standard deviation is 10.

Substituting the value of σ in 3σ = 90 - µ, we get

3×10=90-μ30=90-μμ=90-30μ=60

Therefore, the mean is 60.

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Most popular questions from this chapter

Compare the z-scores to decide which of the following x values lie the greatest distance above the mean and the greatest distance below the mean.

a. x=100,  μ=50,  σ=25b. x=1,  μ=4,  σ=1c. x=0,  μ=200,  σ=100d.x=10,  μ=5,  σ=3

Monitoring weights of flour bags.When it is working properly, a machine that fills 25-pound bags of flour dispenses an average of 25 pounds per fill; the standard deviation of the amount of fill is .1 pound. To monitor the performance of the machine, an inspector weighs the contents of a bag coming off the machine’s conveyor belt every half hour during the day. If the contents of two consecutive bags fall more than 2 standard deviations from the mean (using the mean and standard deviation given above), the filling process is said to be out of control, and the machine is shut down briefly for adjustments. The data

given in the following table are the weights measured by the inspector yesterday. Assume the machine is never shut down for more than 15 minutes at a time. At what times yesterday was the process shut down for adjustment? Justify your answer.

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25.10

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Symmetric or skewed?Would you expect the data sets described below to possess relative frequency distributions that are symmetric, skewed to the right, or skewed to the left? Explain.

a.The salaries of all persons employed by a large university

b.The grades on an easy test

c.The grades on a difficult test

d.The amounts of time students in your class studied last week

e.The ages of automobiles on a used-car lot

f.The amounts of time spent by students on a difficult examination (maximum time is 50 minutes)

Active nuclear power plants.Refer to Exercise 2.54 (p. 98) and the Nuclear Energy Institute’s data on the number of nuclear power plants operating in each of 30 states.

a.Find the range, variance, and standard deviation of this data set.

b.Eliminate the largest value from the data set and repeat part a.What effect does dropping this measurement have on the measures of variation found in part a?

c.Eliminate the smallest and largest value from the data set and repeat part a. What effect does dropping both of these measurements have on the measures of variation found in part a?

State

Status

Number of Power Plants

Alabama

Regulated

2

Arizona

Regulated

1

Arkansas

Regulated

1

California

Regulated

1

Connecticut

Deregulated

1

Florida

Regulated

3

Georgia

Regulated

2

Illinois

Deregulated

6

Iowa

Deregulated

1

Kansas

Regulated

1

Louisiana

Regulated

2

Maryland

Deregulated

1

Massachusetts

Deregulated

1

Michigan

Deregulated

3

Minnesota

Regulated

2

Mississippi

Regulated

1

Missouri

Regulated

1

Nebraska

Regulated

2

New Hampshire

Deregulated

1

New Jersey

Deregulated

3

New York

Deregulated

4

North Carolina

Regulated

3

Ohio

Deregulated

2

Pennsylvania

Deregulated

5

South Carolina

Regulated

4

Tennessee

Regulated

2

Texas

Deregulated

2

Virginia

Regulated

2

Washington

Regulated

1

Wisconsin

Deregulated

1

U.S. business bankruptcies.The American Bankruptcy Institute and the National Bankruptcy Research Center monitor the number of business bankruptcy filings each quarter. The table below lists the number of business bankruptcy filings for each quarter of a recent 2-year period. FILING

a.Explain why the data in the table represent time series data.

b.Construct a time series plot for the quarterly number of bankruptcy filings.

c.Do you detect a trend in the time series plot? Explain.

Year

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2011

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2

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11,705

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2012

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2

3

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10,998

10,374

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9,231
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