Central limit theorem

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/[12]. Therefore we form z = (150-145)/(30/[12]) = .58 and z = (175-145)/(30/[12]) = 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?

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Questions?