Let $X _ { 1 } , \ldots \ldots , X _ { m }$

question 56

Multiple Choice

Let $X _ { 1 } , \ldots \ldots , X _ { m }$ be a random sample from a normal distribution with variance $\sigma _ { 1 } ^ { 2 } , \text { let } Y _ { 1 } , \ldots \ldots , Y _ { n }$ be another random sample (independent of the $\left. X _ { 2 } ^ { \prime } s \right)$ from a normal distribution with variance $\sigma _ { 2 } ^ { 2 } , \text { and } \mathrm { let } S _ { 1 } ^ { 2 } \text { and } S _ { 2 } ^ { 2 }$ denote the two sample variances. Which of the following statements are not true?

• A) The random variable \(F = \left( S _ { 1 } ^ { 2 } / \sigma _ { 1 } ^ { 2 } \right)/ \left( S _ { 2 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } \right)\)
• Has an F distribution with parameters \(v _ { 1 } = m - 1 \text { and } v _ { 2 } = n - 1\)
• B) The random variables \(( m - 1 ) S _ { 1 } ^ { 2 } / σ _ { 1 } ^ { 2 } \text { and } ( n - 1 ) S _ { 2 } ^ { 2 } / σ _ { 2 } ^ { 2 }\)
• Each have a t distribution with m-1 and n-1 degrees of freedom, respectively.
• C) The hypothesis \(H _ { 0 } : σ _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }\)
• Is rejected if the ratio of the sample variances differs by too much from 1.
• D) In testing \(H _ { o } : σ _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } \text { versus } H _ {0} : σ _ { 1 } ^ { 2 } >σ _ { 2 } ^ { 2 } \text {, }\)
• The rejection region for a level \(\alpha \text { test is } f \geq F _ { \alpha , m - 1 n - 1 }\)
• E) All of the above statements are true.

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