Which of the following is NOT correct about the Central Limit Theorem?

The correct answer is Option (3) → A sample size of less than 30 is considered to be sufficient to hold the Central Limit Theorem.

The Central Limit Theorem says that for a large sample the sampling distribution of the sample mean is approximately normal, whatever the shape of the population. A sample of 30 or more is usually taken as large enough. Each option is checked as a statement; the answer is the statement that is false.

Option (1) When the sample size increases, the mean of the sample of data becomes close to the mean of the overall population. — Correct. The standard error \(\sigma/\sqrt{n}\) shrinks as n grows, so the sample mean settles near the population mean.

Option (2) When the sample size increases, the sampling distribution of the mean approaches a normal distribution, regardless of the shape of the parent population. — Correct. This is the statement of the theorem.

Option (3) A sample size of less than 30 is considered to be sufficient to hold the Central Limit Theorem. — Incorrect. A sample of fewer than 30 is treated as a small sample, for which the theorem is not taken to hold.

Option (4) A sample size of 30 or more is considered to be sufficient to hold the Central Limit Theorem. — Correct. This is the usual rule.

CBSE Applied Mathematics (Class XII):
Central Limit Theorem