Question

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 = 0.10 level of significance using a random sample of n = 64 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 = 12.8 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 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.

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.

Use the Distribution tool to help you answer the questions that follow.

Normal Distribution

Mean = 68.0

Standard Deviation = 3.0

6061626364656667686970Mz-2-10z

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 µ 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 67.05.)

If the true value of µ 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 µ = 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?

Do nothing; its current risks are acceptable.

Decrease its sample size.

Decrease α.

Increase its sample size.

Answer 1

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 error, because this branch will be unprofitable.

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µ = 65, σ = 12.8, n = 64

Right tailed critical value, z crit = ABS(NORM.S.INV(0.1)) = 1.282

Z = (X̅ - µ) / (σ/√n)

Reject if Z ≥ 1.282

X̅ ≥ µ + z*(σ/√n)

X̅ ≥ 65 + (1.282) * 12.8/√64

X̅ ≥ 67.0505

The researchers will reject H₀ if the sample mean is greater than 67.0505

The null hypothesis is rejected when the sample mean M is greater than 67.0505. Therefore, the architectural firm will make the decision not to open a branch in Vermont if M is less than or equal to 67.0505.

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If true µ₁ = 67, then type II error, β =

β = P(X̅ < 67.05 | µ₁ = 67)

= P(Z < (67.05-67)/(12.8/√64) )

= P(Z < 0.03)

Using excel function :

= NORM.S.DIST(0.03, 1)

= 0.5126

Suppose that the true value of µ is 67 years. The probability that the architecture firm commits a Type II error is 0.5126.

If the true value of µ is 67 years, the power of the test is 1-0.5126 = 0.4874.

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What should the firm do?

Increase its sample size.