Question

Let \( X_1, X_2, \dots, X_7 \) be a random sample from a normal population with mean 0 and variance \( \theta>0 \). Let \[ K = \frac{X_1^2 + X_2^2 + \dots + X_7^2}{X_1^2 + X_2^2 + \dots + X_7^2}. \] Consider the following statements: \begin{enumerate} \item The statistics \( K \) and \( X_1^2 + X_2^2 + \dots + X_7^2 \) are independent. \item \( \frac{7K}{2} \) has an \( F \)-distribution with 2 and 7 degrees of freedom. \item \( E(K^2) = \frac{8}{63} \). \end{enumerate} Then which of the above statements is/are true?

• A. (I) and (II) only
• B. (I) and (III) only
• C. (II) and (III) only
• D. (I) only

Hint:

In statistical problems involving the ratio of chi-square variables, always check whether the result follows an \( F \)-distribution, which arises from the ratio of two scaled chi-square distributions.

Answer 1

The Correct Option is C

Step 1: Analyzing Statement (I).
The statistic \( K \) involves the sum of squared normal variables \( X_1^2, X_2^2, \dots, X_7^2 \), which are not independent because they are derived from the same underlying sample of normal variables. Thus, \( K \) and the sum of squared variables \( X_1^2 + X_2^2 + \dots + X_7^2 \) are not independent. Hence, statement (I) is false.

Step 2: Analyzing Statement (II).
The statistic \( \frac{7K}{2} \) is a ratio of chi-square variables. The numerator \( 7K \) follows a chi-square distribution with 7 degrees of freedom, and the denominator follows a chi-square distribution with 2 degrees of freedom. The ratio of two scaled chi-square variables follows an \( F \)-distribution with degrees of freedom equal to the numerator and denominator, making statement (II) true.

Step 3: Analyzing Statement (III).
The expected value of \( K^2 \), based on the properties of the chi-square distribution and variance of the sample mean, is known to be \( E(K^2) = \frac{8}{63} \). Hence, statement (III) is true.

Step 4: Conclusion.
Since statements (II) and (III) are true, the correct answer is (C).