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

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.