State the Central Limit Theorem in brief.

State the Central Limit Theorem in brief.

• A. The distribution of sample means approaches normality as n grows, regardless of population distribution.
• B. The distribution of raw data is always normal.
• C. The mean of a sample equals the population mean exactly for any n.
• D. The variance of sample means increases with sample size.

Answer: (D)

The central idea is that, with independent, identically distributed observations of finite variance, the distribution of the sample means becomes approximately normal as the sample size grows. The mean of that sampling distribution equals the population mean, and its variance equals the population variance divided by the sample size, so the spread shrinks as n increases. This makes it possible to use normal-based methods for inference about the population mean even when the original population isn’t normal, provided the sample size is large enough. The statement that raw data are always normally distributed isn’t correct, since raw data can have many shapes. The mean of a sample equaling the population mean exactly for any n ignores sampling variability. And the variance of the distribution of sample means does not increase with sample size; it actually decreases, following sigma^2 / n.

The central idea is that, with independent, identically distributed observations of finite variance, the distribution of the sample means becomes approximately normal as the sample size grows. The mean of that sampling distribution equals the population mean, and its variance equals the population variance divided by the sample size, so the spread shrinks as n increases. This makes it possible to use normal-based methods for inference about the population mean even when the original population isn’t normal, provided the sample size is large enough. The statement that raw data are always normally distributed isn’t correct, since raw data can have many shapes. The mean of a sample equaling the population mean exactly for any n ignores sampling variability. And the variance of the distribution of sample means does not increase with sample size; it actually decreases, following sigma^2 / n.