Question

Let $X_{1}, X_{2}, \dots, X_{n}$ be random sample from Normal population with mean $\mu$ and variance $\sigma^{2}$. Then which of the following results are correct?
A. $\overline{X}\sim N(\mu,\frac{\sigma^{2}}{n})$

B. $\sum_{i=1}^{n}(\frac{X_{i}-\overline{X}}{\sigma})^{2}\sim\chi_{n}^{2}$

C. $\overline{X}$ and $\sum_{i=1}^{n}(\frac{X_{i}-\overline{X}}{\sigma})^{2}$ are independently distributed

D. $\frac{(\overline{X}-\mu)^{2}}{\frac{\sigma^{2}}{n}}\sim \chi_{1}^{2}$

E. $\sum_{i=1}^{n}(\frac{X_{i}-\mu}{\sigma})^{2}\sim \chi_{n-1}^{2}$

• A. A, B only
• B. B, D only
• C. A, C and D only
• D. B, E only

Hint:

Remember: Using the sample mean $\overline{X}$ instead of the population mean $\mu$ in the Chi-square sum "costs" you exactly one degree of freedom. This is why the sum involving $\overline{X}$ has $n-1$ degrees of freedom.

Answer 1

The Correct Option is C

We will evaluate each statement based on the sampling distributions of the mean and variance for a normal population.

Step 1: \color{redEvaluate Statement A
For a sample from $N(\mu, \sigma^2)$, the sample mean $\overline{X}$ is known to be normally distributed with the same mean $\mu$ and variance $\sigma^2/n$
Result: Correct.

Step 2: \color{redEvaluate Statement B
The statistic $\sum_{i=1}^{n}(\frac{X_{i}-\overline{X}}{\sigma})^{2}$ is equivalent to $\frac{(n-1)S^2}{\sigma^2}$. This follows a Chi-square distribution with $(n-1)$ degrees of freedom, not $n$
Result: Incorrect.

Step 3: \color{redEvaluate Statement C
By Basu's Theorem or the properties of the Normal distribution, the sample mean $\overline{X}$ and the sample variance $S^2$ (and by extension its normalized sum of squares) are independent
Result: Correct.

Step 4: \color{redEvaluate Statement D
Since $\overline{X} \sim N(\mu, \sigma^2/n)$, the standardized variable $Z = \frac{\overline{X}-\mu}{\sigma/\sqrt{n}}$ follows $N(0,1)$. The square of a standard normal variable, $Z^2$, follows a $\chi^2_1$ distribution
Result: Correct.

Step 5: \color{redEvaluate Statement E
When the population mean $\mu$ is known, $\sum_{i=1}^{n}(\frac{X_{i}-\mu}{\sigma})^{2}$ is the sum of $n$ independent squared standard normal variables, which follows $\chi^2_n$, not $\chi^2_{n-1}$
Result: Incorrect.

Step 6: \color{redConclusion
Statements A, C, and D are correct.
Final Answer: (3).