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 = 0.01 level of significance using a random sample of n = 169 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 = 22.1 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 Distributions 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 69 years. The probability that the architecture firm commits a Type II error is .

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 = 0.01 level of significance using a random sample of n = 169 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 = 22.1 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 Distributions 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 69 years. The probability that the architecture firm commits a Type II error is .

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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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 = 0.01 level of significance using a random sample of n = 169 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 = 22.1 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 Distributions 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 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 µ 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 µ = 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.

Do nothing; its current risks are acceptable.

Decrease α.

Decrease its sample size.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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