Which of the following statements is true?
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.
For a large enough sample size, the Central Limit Theorem states that the sample medians of repeated samples of a population are normally distributed.
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.
For a large enough sample size, the Central Limit Theorem states that the sample means of repeated samples of a population are normally distributed.
surely you have read the CLT. What does it say?
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.
To understand why this statement is true, let's break it down and explain the Central Limit Theorem. The Central Limit Theorem is a fundamental concept in statistics that applies to sample means or sums. It states that for a large enough sample size, the distribution of the sample means (or sums) will approach a normal distribution, regardless of the shape of the underlying population distribution.
To apply the Central Limit Theorem, there are a few conditions that must be met:
1. A Large Sample Size: The sample size should be sufficiently large. There is no hard and fast rule for what constitutes a "large" sample, but a commonly used guideline is that the sample size should be greater than or equal to 30. As the sample size increases, the sample mean becomes a more reliable estimate of the population mean, and its distribution becomes more closely approximated by a normal distribution.
2. Independence: The observations within the sample should be independent of each other. This means that the value of one observation should not influence the value of another. Independence is important for the Central Limit Theorem to hold.
3. No Finite Standard Deviation: The population that is being sampled should not have a finite standard deviation. This condition is not required for the Central Limit Theorem to hold. In fact, the theorem applies to any population distribution, regardless of whether the standard deviation is finite or not. However, when the population has a finite standard deviation, alternative techniques like the Student's t-distribution may need to be used.
Based on these conditions, the statement "For a large enough sample size, the Central Limit Theorem states that the sample means of repeated samples of a population are normally distributed" is true. This means that if you obtain repeated samples from a population and calculate the means of those samples, the distribution of those sample means will be approximately normal, assuming the sample size is sufficiently large.