# Central limit theorem

• For a population with a mean Î¼ and a variance Ïƒ2 the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed
• The mean of the sampling distribution equal to Î¼ and the variance equal to Ïƒ2/n

As sample size gets large (typically > 30)
Sampling distribution becomes almost normal regardless of shape of population

Example

Suppose the standard deviation of a normal population is known to be 10 and the mean is hypothesized to be 8. Suppose a sample size of 100 is considered. What is the range of sample means that allows the hypothesis to be accepted at a level of significance of 0.05?
Between -11.60 and 27.60
Between 6.04 and 9.96
Between 6.355 and 9.645
Between -8.45 and 24.45

Solution

B.
To accept the hypothesis at a 0.05 significance level, the test statistic Z must fall between -1.96 and 1.96
Z = (X - 8) / (10 / Sqrt (100))  and -1.96 <=z<=1.96
which implies that the sample mean X must be between 6.04 and 9.96