Answer to Question #137897 in Statistics and Probability for Susan Gerea

Question

When is the sample mean normally distributed? (5 marks)

Answer 1

The mean of the sampling distribution always equals the mean of the population. 

$$\mu_{\bar{x}}=\mu$$

The standard deviation of the sampling distribution is σ/√n, where n is the sample size

$$\sigma_{\bar{x}}=\sigma/\sqrt{n}$$

When a variable in a population is normally distributed, the sampling distribution of for all possible samples of size n is also normally distributed. 

If the population is N ( µ, σ) then the sample means distribution is N ( µ, σ/ √ n).

Central Limit Theorem: When randomly sampling from any population with mean µ and standard deviation σ, when n is large enough, the sampling distribution of is approximately normal: ~ N ( µ, σ/ √ n ).

How large a sample size?

It depends on the population distribution. More observations are required if the population distribution is far from normal.

A sample size of 25 is generally enough to obtain a normal sampling distribution from a strong skewness or even mild outliers.

A sample size of 40 will typically be good enough to overcome extreme skewness and outliers.

In many cases, n = 25 isn’t a huge sample. Thus, even for strange population distributions we can assume a normal sampling distribution of the mean and work with it to solve problems.