Central Limit Theorem--statistics

As the sample size increases, the sampling distribution of sample
means approaches a normal distribution

If all  possible random samples of size n are selected from a population with mean μ and standard deviation σ, the mean of the sample means is denoted by μ _x̄ , so
μ _x̄ = μ
the standard deviation of the sample means is:
σ _x̄ = σ⁄ n


Example:  Given that the population of men has normally distributed weights, with a mean of 173 lb and a standard deviation of 30 lb, find the probability that
a. if 1 man is randomly selected, his weight is greater than 180 lb.
b. if 36 different men are randomly selected, their mean weight is greater that 180 lb.


Solution: a) z = (x - μ)/ σ = (180-173)/30 = 0.23
For normal distribution P(Z>0.23) = 0.4090
b) σ _x̄ = σ/n = 20/√ 36 = 5
z= (180-173)/5 = 1.40
P(Z>1.4) = 0.0808