Final answer:
The Central Limit Theorem asserts that with a sufficiently large sample size, the distribution of sample means will be approximately normal, regardless of the population distribution. The claim that the CLT works for any sample size is false, as it requires a sufficiently large sample, often n ≥ 30.
Step-by-step explanation:
The statement regarding the Central Limit Theorem (CLT) needs clarification. The Central Limit Theorem dictates that if you have a sufficiently large sample size, typically n ≥ 30, the distribution of the sample means will approximate a normal distribution regardless of the population's initial distribution shape. However, the theorem does not apply 'no matter what the sample size is' as the student suggests. A smaller sample size may not result in a distribution of sample means that approximates normality.
According to the Central Limit Theorem, for a given large enough sample size drawn from a population, the means of these samples will form a distribution that is approximately normal. This distribution will have the same mean as the population mean (µ), and its standard deviation, known as the standard error, will be the population standard deviation (σ) divided by the square root of the sample size (n). The law of large numbers further supports the theorem by stating that as sample size increases, the sample mean gets closer to the population mean.
For practical purposes, when applying the Central Limit Theorem, one must assure that the sample size is sufficiently large. Therefore, the correct response to the original question is (b) false. A central aspect of the theorem is that it only applies to samples of sufficient size, and it is commonly recommended that this size be at least 30.