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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Definitions:

Prevalence

Prevalence refers to the proportion of a particular population found to be affected by a specific condition at a given time.

Psychological Assessment

The process of testing that uses a combination of techniques to help arrive at some hypotheses about a person and their behavior, personality, and capabilities.

Psychological Testing

The process of administering standardized assessments designed to evaluate an individual's cognitive, emotional, or personality characteristics.

Clinical Significance

An attribute of research results, referring to the practical utility of the treatment studied, which does not follow automatically from the results’ statistical significance.