To determine which statement is true regarding the Central Limit Theorem (CLT), let's analyze the definition and implications of the CLT.
The Central Limit Theorem states:
1. For a large enough sample size, the sample means of repeated samples of a population are normally distributed, regardless of the population's distribution. This is one of the core tenets of the CLT and is fundamental to many statistical methods, as it allows assumptions of normality to be made for the distribution of sample means.
2. Even with a very large sample size, the Central Limit Theorem states that the sample means of repeated samples of a population cannot be normally distributed. This statement is incorrect. The CLT explicitly states that with a sufficiently large sample size, the distribution of the sample means will approach a normal distribution.
3. For a large enough sample size, the Central Limit Theorem states that the sample medians of repeated samples of a population are normally distributed. This statement is incorrect. The CLT specifically applies to the means of the samples, not the medians.
4. For the Central Limit Theorem to be true, you must have a large sample, the underlying population must be normally distributed, and the standard deviation should not be finite. This statement is incorrect for several reasons:
- The underlying population does not need to be normally distributed. The beauty of the CLT is that it applies regardless of the population’s distribution.
- The requirement that the standard deviation should not be finite is incorrect. The population should have a finite standard deviation for the CLT to apply effectively.
Given the analysis, the correct statement is:
For a large enough sample size, the Central Limit Theorem states that the sample means of repeated samples of a population are normally distributed.
