Central limit theorem

• The mean of a sample (x-bar [an overscored lowercase x]) is a random variable, the value of x-bar will depend on which individuals are in the sample.
• E[x-bar] = µ (The expected value of the mean of a sample (x-bar) is equal to the mean of the population (µ).) The law of large numbers says that x-bar will be close to µ for large n (n is the size of the sample). How large n must be is best illustrated by simulation.
• The standard deviation of x-bar (denoted by *sigma* with a subscript x-bar) is equal to *sigma*/(n^.5) (*sigma* is the standard deviation of the population.)
• The central limit theorem says that for large n (sample size), x-bar is approximately normally distributed; the mean is µ and the standard deviation is *sigma*/(n^.5) as noted above. An illustration of the rapidity with which the central limit theorem manifests is illustrated by rolling dice.
N.B.: The above assumes that the sample is randomly drawn from the population.

Example (this should be readable from PC's, may be readable from Macs, and will probably not be readable from unix machines).

If weights are normally distributed with mean m = 145 and standard deviation s = 30, what is the probability that the mean of a sample of twelve weights ([`x]) is between 150 and 175?

[`x] is approximately normally distributed with mean m[`x] = 145 and standard deviation s[`x] = 30/[Ö12]. Therefore we form z = (150-145)/(30/[Ö12]) = .58 and z = (175-145)/(30/[Ö12]) = 3.46. From the normal table, the area to the left of 3.46 is .9997, the area to the le ft of .58 is .7190, hence the area between those two z-scores is .9997-.7190 = . 2807. This is the probability that [`x] is between 150 and 175.

Applets: An applet by R. Todd Ogden illustrates that rolling a single die has the uniiform distribution, but the total number of pips approaches the normal distribution when more dice are rolled. Acoin flipping simulation from University of Alabama at Huntsville illustrates both the convergence to the normal distribution with the number of coins flipped, and the deviation between the observed and expected values.

Competencies: If weight is normally distributed with mean 140 and standard deviation 35 (N(140,35²)), what is the probability that the total weight of 10 individuals is greater than 1500 pounds?