T-Distribution
- We've been talking as if we know what
 is, and therefore that we also know the
population standard error,  . Of course this isn't true in actual
situations. We have to estimate the population variance with the sample variance and
estimate the population standard error with the sample standard error.
- We also don't know the actual population mean. This isn't usually so much of a problem.
Typically we assume a specific value of
 and ask questions such as "Would we
expect this value of  if the population mean were actually ?" See some examples of this here.
- Since we don't know we can't calculate the population z-score. So we calculate a
t-statistic instead:
- The t-statistic has a t-distribution if:
- The population the sample was taken from was normally distributed, and
- The sample was randomly selected
- There are actually many t-distributions with each differing in the number of degrees
of freedom it has. The degrees of freedom, abbreviated df, is n-1. There is a
separate t-distribution table for each value of the degrees of freedom.
For instance, if n=10 then the degree of freedom is 9. To find
P(t > 2 | df = 9) you would use the t-table with the
title df=9. You use these tables just like the z-table for the standard normal
distribution. So, P(t > 2 | df = 9) = 0.0383.
- Regardless of the degrees of freedom, the t-distribution is always symmetric with mean
0. So all the tricks you used to find probabilities with the z-table also work with the
t-tables. For instance:
P(t < -2 | df = 9) =
P(t > 2 | df = 9) =
0.0383
- As n becomes large, and therefore df gets large, the t-distribution table becomes more
and more similar to the z-distribution table. Indeed, the tables I provided only go up to
df=30. If you are ever confronted with a df bigger than 30 just use the standard normal
table (= z-distribution table).
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