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

The Correct Option is A, B, D

Analysis of the Given Statements on Sample Mean and Distribution

Option (A): The Sample Mean has Standard Deviation √(σ/n)

• Correct. The standard deviation of the sample mean is given by: 
• This follows from the property that the variance of the sample mean is:

Option (B): The Probability Distribution of the Standardized Sum Tends to Normal as n → ∞

• Correct. By the Central Limit Theorem (CLT), as n → ∞, the sampling distribution of the standardized sum:
• approaches a standard normal distribution N(0,1), regardless of the shape of the original population distribution, provided σ² is finite.

Option (C): (n−1)S² / σ² follows a χ² Distribution with (n−1) Degrees of Freedom

• Incorrect. The given statistic:
• follows a χ² distribution with (n−1) degrees of freedom only if the underlying population is normally distributed.
• If the population is not normally distributed, this result does not hold.

Option (D): The Sample Mean is Always a Consistent Estimator of μ

• Correct. Consistency means that the estimator converges in probability to the true parameter as n → ∞.
• The sample mean X̄ is an unbiased estimator of μ, and as n → ∞, the variance of X̄:
• approaches 0, ensuring consistency.

Final Answer

The correct options are:

• (A) The sample mean has standard deviation √(σ/n).
• (B) The standardized sum follows a standard normal distribution as n → ∞.
• (D) The sample mean is a consistent estimator of μ.