Which of the following are correct?(A) The central limit theorem (CLT…

The correct answer is Option (4) → (A), (B), (C) and (D)

A population is the whole group under study, and a measure that describes it (such as μ or σ) is a parameter. A sample is a part of the population, and a measure that describes the sample is a statistic. The central limit theorem says that, for samples of size n from any population with mean μ and standard deviation σ, the sampling distribution of the sample mean \( \overline{x} \) approaches a normal distribution with mean μ and standard deviation \( \frac{\sigma}{\sqrt{n}} \) as n grows, whatever the shape of the population.

So a parameter describes the population, a study of the population identifies its characteristics, and a sample is used to draw inferences about the population.

Checking the statements

(A) The central limit theorem (CLT) states that the sampling distribution of the sample approaches a normal distribution as the sample size gets larger, no matter what is the shape of the population distribution. — Correct. This is the central limit theorem for the sample mean.

(B) The measurable characteristic of population is called parameter. — Correct. A parameter describes the population; the matching measure of a sample is a statistic.

(C) The population focuses on the identification of the characteristic. — Correct. A study of the population aims to identify its characteristics.

(D) The sample makes inference about the population. — Correct. A sample is studied in order to draw inferences about the population it comes from.

Note: It seems that there is a mistake in the original NTA paper, which prints “the sampling distribution of the sample” in (A) when it should read “the sampling distribution of the sample mean”. The question can still be solved because the rest of the statement is the standard form of the central limit theorem.