The t, , and F Distributions

The same idea can be applied to all of the transformed statistics except that only the ones labeled Z follow the standard normal distribution. To study the rest, we need to understand three other distributions known as the t, , and F distributions:

1. While there is only one Z curve, there are actually infinitely many t, , and F curves.
2. For the t and curves, there is one curve for each value of a positive integer (1, 2, 3, and so on) called degrees of freedom (see the column labeled df in the table of transformed statistics above). The values on the horizontal axis of the t and curves having degrees of freedom v and area to their right are denoted by and , respectively.
3. The F curves are actually indexed by a pair of positive integers (called numerator and denominator degrees of freedom--see the table above). The value of the F curve having numerator degrees of freedom and denominator degrees of freedom and having area to its right is denoted by .
4. The t curve for increasing degrees of freedom gets closer and closer to the Z curve (see the ``Z, t, Chi-square, F'' concept lab).
5. While Z and t can be negative, the and F are always positive.

One last example. You will see later how to use Stataquest to find

and thus the proportion of samples of size 21 from a normal population having between 9.5911 and 34.17 is 95%, that is,

or dividing all three terms by 20 gives that 95% of all samples of size 21 will have between .4295 and 1.7085. This is our answer to the question ``how close is the sample variance to the population variance''.