OVERVIEW: This section contains one ofthe most important of all statistical theorems, the Central LimitTheorem of Statistics. It also emphasizes that it is conventional theGreek letters m
and s are used for the population parameters mean and standarddeviation, and that x(bar) and s conventionally represent the meanand standard deviation for samples.
THE CENTRAL LIMIT THEOREM
Consider an SRS of size n from any population withmean mand standard deviation s . When n is large, thesampling distribution of x(bar) has the following properties:
(a) it is approximately normal.
(b) the mean of the distribution is
m . (c) the standard deviation of the distribution iss
/sqrt(n).
Here is an example illustrating (b) and (c) of theCentral Limit Theorem.
Consider the population P ={2,4,6}.
For this population, m = 4and s = sqrt[((2-4)^{2} + (4-4)^{2} + (6-4)^{2})/3] = 1.632993162Now, consider all possible samples of size 2 (withreplacement). There would be 3^{2} = 9 such samples.
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The collection of sample means is M ={2,3,3,4,4,4,5,5,6}.
The mean of M =4 = m,illustrating (b) of the Central Limit Theorem.The standard deviationof M = 1.154700538 =
s/ sqrt(2) =1.632993162/ Sqrt(2) ,illustrating (c) of the Central Limit Theorem.
Example:
A tire manufacturer advertised that a new brand of tires had a meanlife of 40,000 miles with a standard deviation of 2,000 miles. Aresearch team examined a random sample of 100 of these tires anddetermined that the tires in the sample had a mean life of 39,000miles. If the mean life is indeed 40,000 miles, how likely is it thata random sample of 100 would have a mean life of 39,000 miles?
Considering the set W of all sample means of size100, the mean of W is 40,000, and the standard deviation of W is2000/sqrt(100) = 200. The probability of getting a sample with a meanof 39,000 or less is