Question

Suppose that many random samples of size n for a variable are taken and the distribution of means of each sample is recorded. Select all statements that are part of the Central Limit Theorem below.

A. The mean of the distribution of means approa the population mean.
B. The stardard deviation of the distribution of means approaches a/ yn, where a is the standard deviation of the population.
C. The standard deviation of the distribution of means approaches a, where a is the standard deviation of the population.
D. The mean of the distribution of means approaches the population mean, p/ yn.
E. The distribution of means will approximately follow the distribution of the variable being sampled.
F. The distribution of means will be approximately a normal distribution

Answer 1

The correct statements of the Central Limit Theorem are that the mean of the distribution of means approaches the population mean, the standard deviation of the distribution of means approaches the population standard deviation divided by the square root of the sample size, and the distribution of means will be approximately normal.

Step-by-step explanation:

The Central Limit Theorem is a fundamental theory in statistics that provides important insights into the distribution of sample means. According to this theorem:

• A. The mean of the distribution of means approaches the population mean (Correct).
• B. The standard deviation of the distribution of means approaches a/√n, where a is the standard deviation of the population (Correct).
• F. The distribution of means will be approximately a normal distribution (Correct).

The incorrect statements are C and D, which contains typographical errors, and E, which falsely asserts that the distribution of means will follow the distribution of the variable being sampled. However, regardless of the original distribution of the variable, the distribution of sample means will follow a normal distribution given a sufficiently large sample size.