"I shall persevere until I findsomething that is certain - or, at least until I find for certainthat nothing is certain."

Rene Descartes (1596 - 1650)

9.3 SAMPLE MEANS (Pages 481 - 494)

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] = 1.632993162

Now, consider all possible samples of size 2 (withreplacement). There would be 32 = 9 such samples.

 Sample Sample Mean 2,2 2 2,4 3 2,6 4 4,2 3 4,4 4 4,6 5 6,2 4 6,4 5 6,6 6

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 normalcdf(-1E99, 39000, 40000,200) = .000000287. In other words, it would be very unlikely to get asample of 100 with a mean of 39,000 miles if the manufacturers claimis true.