Sampling Distributions

Constructing a Sampling Distribution

draw lots of random samples (n of them).
calculate the mean for each sample.
plot the frequency distribution of all these means.
The mean should be mu for the population. The bigger n, the closer to mu.
The sem should be sigma/sqrt(n). The bigger n, the smaller the sem.

Central Limit Theorem

There are 3 properties of the sampling distribution of the mean (for lots and lots of sample means):

It is normal (68% area under curve lies within + and - one sem of mu: not within one sigma of mu)
Xbar = population mean mu (it tends toward mu in the limit of large n)
SEM (standard error of the mean) s xbar = sigma/sqrt(n)
This is the same as saying s xbar = sqrt(sigma**2/n), which we will encounter later when we do parametric tests.

Confidence Intervals

Take some sample mean xbar. A 68% confidence interval around it means xbar + or - one sem; a 95% confidence interval is two sem. That means there is a 68% or 95% chance that the population mean mu lies within this confidence interval. For other values, like 90%, you need to go to a z table and figure how much of the curve is between mu and z (split the difference, it is + or - .45, so z = 1.65 sem's on each side).

If mu lies within that confidence interval, then it is a hit.

General Stuff

The sampling error is xbar - mu.
Mu is a parameter; xbar is a statistic.
The bigger n, the smaller the range of values for the means (it is always smaller than the range of the underlying individual scores).
Likewise, sem is always smaller than sigma.
Increasing the level of a confidence interval increases the probability of including mu within its limits
If the hzero is that xbar = mu, and you do not reject hzero, then you could possibly make a type II error. If you did, you could possibly make a type I error.

Back to Statistics course

Lorraine Sherry
lsherry@carbon.cudenver.edu
Updated October 7, 1996