What is the difference between the t and z distributions? Why is the t used for smaller samples? Is that just the way it is or is there a reason for it?
THANKS VERY MUCH!!!
|A little info|
When used for sample means, the z-distribution assumes that you know the POPULATION standard deviation (which is never the case).
The t-distribution is based on using the sample standard deviation as an estimate of the population standard deviation. Approximating the population standard deviation with a sample standard deviation means the sampling distribution is going to have more spread and is going to be affected by the sample size.
Hence for any test statistic value (t=+/-2 vs z=+/-2) there will be a higher proportion outside those endpoints for the t-distribution.
However, as the sample size gets larger the t-distribution approximates the z-distribution. For sample sizes larger than ~25, the difference between the two is usually negligible
z distribution is exact, t distribution is an approximation because you are using a sample to represent the entire population. The bigger the sample size, the closer the t distribution will be to the real distribution.
t is really the "student t" distribution. "z" is not a distribution but when working with the normal distribution it is relatively common to use notation where the variate "z" has a normal distribution (and often normalized, with a mean of zero and a standard deviation of 1). Look at the t value for a given level of "significance" as the number of degrees of freedom increase. You will notice that as the degrees of freedom increase the t-distribution approaches the normal (as stated by others above).
Another way to look at it is that when you have a small sample size it is easier to actually have no true effect yet have a measured effect of some size. The higher value of the t statistic for the case of a small sample size compensates for it being easier to get a spurious sample value.
|A little info|
Try this website:
It has an applet that lets you change the degrees of freedom (df = n-1). It might help.
Note: Both the z-distribution and t-distribution are theoretical distributions and therefore both are "exact". They are used to model sampling distributions but are only approximations to any reality.
Statistics definitely comes easier to some than to others. If you are having trouble with this, I suggest finding some patient, kind soul in your class that might be willing to spend a few moments with you. While confusing to you right now, they are actually pretty basic. You will find several peers who could probably get you on the right track quickly.
Both are theoretical.
Both are exact.
"...only approximations to any reality."
t-distribution: A distribution which is used largely for inferences concerning concerning the mean (or means) of normal distributions whose variances are unknown.
Uppercase Z is often called the 'standard normal random variable' with lowercase z being the number of standard deviations separating a (test) value from the mean.
Back to you.