Applied Statistics in Business and Economics
Applied Statistics in Business and Economics
5th Edition
ISBN: 9780077837303
Author: David Doane, Lori Seward Senior Instructor of Operations Management
Publisher: McGraw-Hill Education
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Chapter 9, Problem 107CE

a.

To determine

Find the value of β risk if the true means are 495, 490, 485 and 480.

a.

Expert Solution
Check Mark

Answer to Problem 107CE

The β risk values are tabulated below:

True meanβ risk
4950.7915
4900.4913
4850.1962
4800.0457

Explanation of Solution

Calculation:

The given information is that the sample of size 4 are considered. The know standard deviation is 12. Here, the researcher interested to test the hypothesis that μ=500.

The test hypotheses are given below:

Null hypothesis:

H0:μ500.

Alternative hypothesis:

H1:μ<500.

If the true mean is 495, the value of β risk is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 4, Differences as –5, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  1

From the output, the power is 0.2085.

The value of β risk is,

β=1Power=10.2085=0.7915

Thus, the value of β risk is 0.7915.

If the true mean is 490, the value of β risk is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 4, Differences as –10, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  2

From the output, the power is 0.5087.

The value of β risk is,

β=1Power=10.5087=0.4913

Thus, the value of β risk is 0.4913.

If the true mean is 485, the value of β risk is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 4, Differences as –15, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  3

From the output, the power is 0.8038.

The value of β risk is,

β=1Power=10.8038=0.1962

Thus, the value of β risk is 0.1962.

If the true mean is 480, the value of β risk is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 4, Differences as –20, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  4

From the output, the power is 0.9543.

The value of β risk is,

β=1Power=10.9543=0.0457

Thus, the value of β risk is 0.0457.

b.

To determine

Find the power if the true means are 495, 490, 485 and 480 and sketch the power curve.

b.

Expert Solution
Check Mark

Answer to Problem 107CE

The power values are tabulated below:

True meanPower
4950.2085
4900.5087
4850.8038
4800.9543
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  5

Explanation of Solution

Calculation:

If the true mean is 495, the power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 4, Differences as –5, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  6

From the output, the power is 0.2085.

If the true mean is 490, the power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 4, Differences as –10, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  7

From the output, the power is 0.5087.

If the true mean is 485, the power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 4, Differences as –15, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  8

From the output, the power is 0.8038.

If the true mean is 480, the power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 4, Differences as –20, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  9

From the output, the power is 0.9543.

Sketch the power curve.

Software procedure:

Step-by-step software procedure to construct the power curve using MINITAB is as follows:

  • Choose Graph > Scatterplot.
  • Choose With Connect Line, and then click OK.
  • Under Y variables, enter Power.
  • Under X variables, enter True mean.
  • Click OK.

c.

To determine

Find the β risk and power using n=16 and also sketch the power curve.

c.

Expert Solution
Check Mark

Answer to Problem 107CE

The β risk and power values are tabulated below:

True meanPowerβ risk
4950.50870.4913
4900.95430.0457
4850.99960.0004
48010
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  10

Explanation of Solution

Calculation:

The given information is that the sample of size 16 are considered. The know standard deviation is 12. Here, the researcher interested to test the hypothesis that μ=500.

The true means are 495, 490, 485 and 480.

If the true mean is 495, the β risk and power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 16, Differences as –5, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  11

From the output, the power is 0.5087.

The value of β risk is,

β=1Power=10.5087=0.4913

Thus, the value of β risk is 0.4913.

If the true mean is 490, the β risk and power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 16, Differences as –10, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  12

From the output, the power is 0.9543.

The value of β risk is,

β=1Power=10.9543=0.0457

Thus, the value of β risk is 0.0457.

If the true mean is 485, the β risk and power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 16, Differences as –15, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  13

From the output, the power is 0.9996.

The value of β risk is,

β=1Power=10.9996=0.0004

Thus, the value of β risk is 0.0004.

If the true mean is 480, the β risk and power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

  • Select Stat > Power and sample size > 1 sample z.
  • Enter Sample sizes as 16, Differences as –20, and Standard deviation as 12.
  • Click OK.
  • Output using MINITAB software is given below:

Applied Statistics in Business and Economics, Chapter 9, Problem 107CE , additional homework tip  14

From the output, the power is 1.

The value of β risk is,

β=1Power=11=0

Thus, the value of β risk is 0.

Sketch the power curve.

Software procedure:

Step-by-step software procedure to construct the power curve using MINITAB is as follows:

  • Choose Graph > Scatterplot.
  • Choose With Connect Line, and then click OK.
  • Under Y variables, enter Power.
  • Under X variables, enter True mean.
  • Click OK.

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Chapter 9 Solutions

Applied Statistics in Business and Economics

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