Which of the following are correct?
(A) As the sample size gets larger, the sampling distribution of the sample mean approaches a normal distribution, whatever be the shape of the population distribution.
(B) The central limit theorem holds only when the population itself is normally distributed.
(C) The sampling distribution of a statistic is the distribution of the values of that statistic over all possible samples of the same size.
(D) For a fixed population, the standard error of the mean increases as the sample size increases.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
The sampling distribution of a statistic is made of its values over all possible samples of one fixed size. The central limit theorem says that, for large samples, the sampling distribution of the mean is close to normal for any shape of population.
The standard error of the mean is
\( \displaystyle \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \)
which gets smaller as n gets larger.
So the CLT needs no normal population, and the standard error falls as the sample size rises.
Checking the statements
(A) As the sample size gets larger, the sampling distribution of the sample mean approaches a normal distribution, whatever be the shape of the population distribution. — Correct. This is the central limit theorem.
(B) The central limit theorem holds only when the population itself is normally distributed. — Incorrect. It holds for any shape of population.
(C) The sampling distribution of a statistic is the distribution of the values of that statistic over all possible samples of the same size. — Correct. This is its definition.
(D) For a fixed population, the standard error of the mean increases as the sample size increases. — Incorrect. \(\sigma/\sqrt{n}\) decreases when n increases.