Which of the following statements are correct about the “Central Limit Theorem”?
(A) The sampling distribution of the sample mean approaches the normal distribution as the sample size gets larger.
(B) A sample size of 30 or more is considered to be sufficient to hold the “Central Limit Theorem”.
(C) As the sample size becomes larger, the prediction of characteristics of the population becomes more accurate.
(D) The sampling distribution of the sample mean approaches a bell shaped curve as the sample size gets larger.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B), (C) and (D)
The Central Limit Theorem says that for a large sample, the sampling distribution of the sample mean \(\bar{x}\) is approximately normal, with mean \(\mu\) and standard error \(\frac{\sigma}{\sqrt{n}}\), whatever the shape of the population.
(A) The sampling distribution of the sample mean approaches the normal distribution as the sample size gets larger. — Correct. This is the statement of the theorem.
(B) A sample size of 30 or more is considered to be sufficient to hold the “Central Limit Theorem”. — Correct. A sample with n ≥ 30 is the usual rule for a “large” sample.
(C) As the sample size becomes larger, the prediction of characteristics of the population becomes more accurate. — Correct. The standard error \(\frac{\sigma}{\sqrt{n}}\) falls as n grows, so the sample means lie closer to the population mean, and estimates of the population characteristics become more accurate.
(D) The sampling distribution of the sample mean approaches a bell shaped curve as the sample size gets larger. — Correct. The normal curve is bell shaped, so this restates the theorem.
CBSE Applied Mathematics (Class XII):
Central Limit Theorem (CLT)
