Question

Which of the following statements is not true about the central limit theorem?

1) For a large enough sample size, the central limit theorem states that the sample means of repeated samples of a population are normally distributed.
2) The central limit theorem is generally applicable only for data sets comprising 30 or more samples.
3) To apply the central limit theorem to a sampling distribution, the underlying population (the population sampled) must also be normally distributed.
4) According to the central limit theorem, the mean of the sampling distribution is equal to the population mean.

Answer 1

Statement number 3 is incorrect regarding the Central Limit Theorem; the CLT applies even when the underlying population is not normally distributed. The CLT ensures that sample means will form a normal distribution as the sample size increases, regardless of the population's distribution.

Step-by-step explanation:

The student asked which statement about the Central Limit Theorem (CLT) is not true. The incorrect statement is number 3: To apply the central limit theorem to a sampling distribution, the underlying population (the population sampled) must also be normally distributed. This is not true because the CLT states that the distribution of sample means will be normal, regardless of the population's distribution, provided the sample size is large enough.

The CLT actually allows for the underlying population to have any distribution. It states that as the sample size increases, the sampling distribution of the sample means will approach a normal distribution (normality). This applies even if the original population is not normally distributed. The sample size of 30 is often considered sufficiently large for the CLT to hold, but it's not a strict cutoff. The other statements regarding the CLT are accurate, including that the mean of the sampling distribution is equal to the population mean, reflecting the alignment with the law of large numbers.