Four Independent Samples of 100 Values Each Are Randomly Drawn $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 }$

question 45

Essay

Four independent samples of 100 values each are randomly drawn from populations that are normally distributed with equal variances. You wish to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 }$ $= \mu _ { 4 }$
i) If you test the individual claims $\mu _ { 1 } = \mu _ { 2 } , \mu _ { 1 } = \mu _ { 3 } , \mu _ { 1 } = \mu _ { 4 } , \ldots , \mu _ { 3 } = \mu _ { 4 }$ , how many ways can you pair off the 4 means?
ii) Assume that the tests are independent and that for each test of equality between two means, there is a $0.99$ probability of not making a type I error. If all possible pairs of means are tested for equality, what is the probability of making no type I errors?
iii) If you use analysis of variance to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 }$ at the $0.01$ level of significance, what is the probability of not making a type I error?

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Definitions:

Indices

Statistical measures that track changes in a representative group of individual data points, often used to assess economic and financial performance.

Price Support

A government policy or mechanism that sets a minimum price for a product or commodity to help stabilize the market and ensure income for producers.

Taxpayers

Individuals or entities that are obligated to pay taxes to a government authority, based on earnings, property ownership, or consumption.

Farmers

Individuals or entities engaged in the practice of agricultural production, cultivating plants and livestock for food, fiber, and other products.