Statistical Analysis - z and t StatisticsDate: 07/06/2004 at 13:32:52 From: Carl Subject: Research Applications/Statistics Dr. Math, Could you tell me the differences between z-statistics and t- statistics? Date: 07/06/2004 at 14:53:25 From: Doctor Mitteldorf Subject: Re: Research Applications/Statistics Hi Carl - Suppose you have a large population, and you want to determine the mean of that large population. It is impractical to measure all of them and take the average longhand, because there are so many. The standard approach is sampling, which means you select n of them at random, and take the average of those. The first question you ask is: what's my best estimate of the average of the entire population, and the answer is, as you expect, that the average of your sample of n is your best estimate of the average of the entire population. The second question you want to ask is: after measuring a sample of n, how well do I know the average of the entire population? The answer as a tolerance, or "plus or minus" is the "standard deviation of the mean". It is standard deviation of your sample of n, divided by the square root of (n-1). tolerance in estimate of population average = std dev of mean = sqrt((<x^2> - <x>^2)/(n-1)) The third question you may want to ask--a deeper level of detail--is a full probability distribution for estimating the population average. In other words, given your sample with its mean m and standard deviation s, can you assign a probability that the population mean is equal to any arbitrary value mu? It is here that the Student-t distribution comes into play. In many situations you may have reason to believe that the large population is Normally distributed (what you have called z statistics). If so, then mu is t-distributed, with a mean of m and a standard deviation equal to the standard deviation of mean calculated above. The curve for the t distribution has the bell shape reminiscent of a Normal distribution. But the t distribution is not just one shape, but a family of shapes dependent on n. For small n, the tails of the distribution are much longer than the tails of a Normal distribution, reflecting a broad uncertainty of the value of the true population mean. For large n, the t distribution approaches the Normal distribution as a limit. In practice, a rule of thumb says that if your n is greater than 50, you needn't bother with the t-distribution, but can use the Normal distribution as an approximation. You can read more about the t distribution in stat text books, and at many web sites. Here is a sample: The Student t Distribution http://www.fmi.uni-sofia.bg/vesta/Virtual_Labs/special0/special4.htm Student's t Distribution http://mathworld.wolfram.com/Studentst-Distribution.html Sampling http://astro.temple.edu/~mjoshua/Assgn-6.doc Statistical Distributions - Student t Distribution - Example http://www.xycoon.com/studentt.htm - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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