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: Find the distribution of \(\bar{X}\).
Since \(X_1,X_2,\ldots,X_9\) are from \(N(\mu,\sigma^2)\), the sample mean is
\[ \bar{X}\sim N\left(\mu,\frac{\sigma^2}{9}\right) \]
Also,
\[ Y\sim N(\mu,\sigma^2) \]

Step 2: Find the distribution of \(\bar{X}-Y\).
Since \(\bar{X}\) and \(Y\) are independent,
\[ Var(\bar{X}-Y)=Var(\bar{X})+Var(Y) \]
\[ Var(\bar{X}-Y)=\frac{\sigma^2}{9}+\sigma^2 \] \[ Var(\bar{X}-Y)=\frac{10\sigma^2}{9} \]
Also,
\[ E(\bar{X}-Y)=\mu-\mu=0 \]
Therefore,
\[ \bar{X}-Y\sim N\left(0,\frac{10\sigma^2}{9}\right) \]

Step 3: Standardize \(\bar{X}-Y\).
Hence,
\[ \frac{\bar{X}-Y}{\sigma\sqrt{\frac{10}{9}}}\sim N(0,1) \]
Since
\[ \sqrt{\frac{10}{9}}=\frac{\sqrt{10}}{3}, \] we get
\[ \frac{3}{\sqrt{10}}\left(\frac{\bar{X}-Y}{\sigma}\right)\sim N(0,1) \]

Step 4: Use the distribution of sample variance.
For a normal sample of size \(9\),
\[ \frac{(n-1)S^2}{\sigma^2}\sim \chi_{n-1}^2 \]
Here, \(n=9\), so
\[ \frac{8S^2}{\sigma^2}\sim \chi_8^2 \]
Also, \(\bar{X}\) and \(S^2\) are independent for a normal sample, and \(Y\) is independent of the sample.
Therefore, \(\bar{X}-Y\) is independent of \(S^2\).

Step 5: Form the \(t\)-statistic.
The expression is
\[ \frac{3}{\sqrt{10}}\left(\frac{\bar{X}-Y}{S}\right) \]
This can be written as
\[ \frac{\frac{3}{\sqrt{10}}\left(\frac{\bar{X}-Y}{\sigma}\right)} {\frac{S}{\sigma}} \]
The numerator follows standard normal distribution, and the denominator is based on an independent chi-square variable with \(8\) degrees of freedom.
Therefore, the statistic follows a Student's \(t\)-distribution with \(8\) degrees of freedom.

Step 6: Final conclusion.
Hence,
\[ \frac{3}{\sqrt{10}}\left(\frac{\bar{X}-Y}{S}\right)\sim t_8 \]
Therefore, the correct answer is
\[ \boxed{t_8} \]