Question

Let $X_1, X_2, … , X_𝑛$ be a random sample of size n > 1 drawn from a probability distribution having mean $\mu$ and non-zero variance $\sigma^2$. Then, which of the following is/are CORRECT?

• A. The sample mean has standard deviation $\sigma/ \sqrt{n}$
• B. The probability distribution of $^nΣ_{𝑖=1} (X_𝑖 − \mu) / \sigma \sqrt{n}$ will tend to follow standard normal distribution as $n→ \infty$
• C. $\frac{(n − 1) S^2}{ \sigma^2}$ will follow $X^2$ distribution with (n − 1) degrees of freedom, where $S^2$ is the sample variance
• D. The sample mean is always a consistent estimator of $\mu$

Answer 1

Answer: (A), (B), (D)

The sample mean has standard deviation $\sigma/ \sqrt{n}$

Answer 2

The correct Options are A and B and D : The sample mean has standard deviation $\sigma/ \sqrt{n}$AND The probability distribution of $^nΣ_{𝑖=1} (X_𝑖 − \mu) / \sigma \sqrt{n}$ will tend to follow standard normal distribution as $n→ \infty$ ANDThe sample mean is always a consistent estimator of $\mu$