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 = .01 level of significance using a random sample of n = 196 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 = 23.8 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 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 error, because this branch will be unprofitable. 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 error, because this branch would have been profitable.
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 = .01 level of significance using a random sample of n = 196 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 = 23.8 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 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 error, because this branch will be unprofitable. 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 error, because this branch would have been profitable.
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![**12. 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 α = .01 level of significance using a random sample of n = 196 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 σ = 23.8 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.
[Chart]
- 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 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 _________ error, because this branch will be unprofitable.
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 _________ error, because this branch would have been profitable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff611932b-28af-4bbe-89f9-c5ee05184fbc%2F86bdad40-5648-4381-9799-d6cf33668104%2Fodr6oml_processed.png&w=3840&q=75)
Transcribed Image Text:**12. 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 α = .01 level of significance using a random sample of n = 196 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 σ = 23.8 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.
[Chart]
- 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 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 _________ error, because this branch will be unprofitable.
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 _________ error, because this branch would have been profitable.
Transcribed Image Text:The null hypothesis is rejected when the sample mean \( M \) is ______________. Therefore, the architectural firm will make the decision not to open a branch in Vermont if \( M \) is ______________.
*(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 \( \mu \) is 69 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 (69) and position the vertical line at the critical value of \( M \) 68.95.)*
If the true value of \( \mu \) is 69 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 \( \mu = 69 \). (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?
- Increase its sample size.
- Decrease \( \alpha \).
- Decrease its sample size.
- Do nothing; its current risks are acceptable.
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