The Central Limit Theorem (CLT) is a theorem in probability theory that states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution.
The CLT is mostly useful as an approximation because it allows us to make inferences about the population mean based on a sample mean. For example, if we know that the population mean is 100 and the standard deviation is 10, and we take a sample of 100 observations, we can use the CLT to calculate the probability that the sample mean is between 90 and 110.
However, the CLT is not always accurate. For example, if the underlying distribution is highly skewed, the CLT may not be a good approximation. Additionally, the CLT only applies to sufficiently large samples. If the sample size is too small, the CLT may not be accurate.
Overall, the CLT is a powerful tool that can be used to make inferences about the population mean based on a sample mean. However, it is important to keep in mind that the CLT is only an approximation, and it may not be accurate for small samples or highly skewed distributions.
Here is a brief explanation of each option:
- Option A: The CLT is mostly useful as an approximation. This is true because the CLT only applies to sufficiently large samples. If the sample size is too small, the CLT may not be accurate.
- Option B: The CLT is not useful as an approximation. This is false because the CLT is a powerful tool that can be used to make inferences about the population mean based on a sample mean.
I hope this helps! Let me know if you have any other questions.