Question

Let \(X_1,X_2,\ldots,X_9,Y\) be independent and identically distributed \(N(\mu,\sigma^2)\) random variables. Define

• A. \(\chi_8^2\)
• B. \(\chi_9^2\)
• C. \(t_8\)
• D. \(t_9\)

Hint:

A statistic has a \(t\)-distribution when a standard normal variable is divided by the square root of an independent chi-square variable divided by its degrees of freedom.

Answer 1

The Correct Option is C

Step 1: Set up the pieces.
From $X_1,\dots,X_9,Y\sim N(\mu,\sigma^2)$, the mean $\bar X\sim N(\mu,\sigma^2/9)$ and $Y$ is on its own.

Step 2: Look at the difference.
Since $\bar X$ and $Y$ are independent, $\bar X-Y$ is normal with mean $0$ and variance $\frac{\sigma^2}{9}+\sigma^2=\frac{10\sigma^2}{9}$. Standardising gives a $N(0,1)$ in the numerator.

Step 3: Bring in $S^2$.
For the nine $X$'s, $\dfrac{8S^2}{\sigma^2}\sim\chi^2_8$, and it is independent of both $\bar X$ and $Y$.

Step 4: Form the ratio.
A standard normal over $\sqrt{\chi^2_8/8}$ is a $t$ with $8$ degrees of freedom. The given statistic is exactly this once $\sigma$ cancels.

Step 5: Conclude.
\[ \boxed{t_8} \]