Suppose an architectural firm specializing in the structural restoration and renovation of historic homes and early barns is deciding whether to open a branch of the company in Wilmington, Vermont. Market research commissioned by the firm indicates that the Vermont location will be profitable only if the mean age of houses and barns located within a 100-mile radius of Wilmington is greater than 65 years. The architectural firm conducts a hypothesis test to determine whether u, the mean age of structures located within a 100-mile radius of Wilmington.
Suppose an architectural firm specializing in the structural restoration and renovation of historic homes and early barns is deciding whether to open a branch of the company in Wilmington, Vermont. Market research commissioned by the firm indicates that the Vermont location will be profitable only if the mean age of houses and barns located within a 100-mile radius of Wilmington is greater than 65 years. The architectural firm conducts a hypothesis test to determine whether u, the mean age of structures located within a 100-mile radius of Wilmington.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section: Chapter Questions
Problem 8SGR
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Transcribed Image Text:9. Computing power
Suppose an architectural firm specializing in the structural restoration and renovation of historic homes and early barns is deciding whether to open a
branch of the company in Wilmington, Vermont. Market research commissioned by the firm indicates that the Vermont location will be profitable only if
the mean age of houses and barns located within a 100-mile radius of Wilmington is greater than 65 years.
The architectural firm conducts a hypothesis test to determine whether µ, the mean age of structures located within a 100-mile radius of Wilmington,
is greater than 65 years. The test is conducted at a = 0.05 level of significance using a random sample of n = 81 houses and barns located in the
specified area. The population standard deviation of the age of structures is assumed to be known with a value of o = 12.6 years. The firm will open a
Vermont branch only if it rejects the null hypothesis that the mean age of structures in the specified area is less than or equal to 65 years.
To summarize this hypothesis test refer to the chart given below.
Reject the Null = Open the Branch
Fail to Reject = Do Not Open the Branch
If, on the basis of the hypothesis test, the architectural firm decides not to open a branch, but the true mean age of houses and barns located within a
100-mile radius of Wilmington is greater than 65 years, the firm has committed a Type II ▼ error, because this branch would have been profitable.
If, on the basis of the hypothesis test, the architectural firm decides to open a branch, but the true mean age of houses and barns located within a
100-mile radius of Wilmington is less than 65 years, the firm has committed a Type I v error, because this branch will be unprofitable.
Transcribed Image Text:The null hypothesis is rejected when the sample mean M
greater than or equal to 67.30
. Therefore, the architectural firm will make the
decision not to open a branch in Vermont if M is
less than 67.30
(Hint: To use the tool, set the Mean to the hypothesized mean (65) and the Standard Deviation to the standard deviation of the sampling distribution
of M.)
Suppose that the true value of u is 67 years. The probability that the architecture firm commits a Type II error is
(Hint: To use the tool, set the Mean to the true mean (67) and position the vertical line at the critical value of M 67.30.)
If the true value of u is 67 years, the power of the test is
Based on the level of significance it has selected for its test, the architectural firm is willing to risk a
probability of opening an
unprofitable branch. It is willing to risk a 0.20 probability of not opening the branch when u = 67. (In other words, it is willing to fail to reject the null
hypothesis when it actually should have rejected the null hypothesis only 20% of the time.) What should the firm do?
O Increase its sample size.
O Decrease a.
O Do nothing; its current risks are acceptable.
O Decrease its sample size.
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