Student t Distribution

Consider some population where the numeric measurement of interest has mean m and standard deviation s. If these measurements are Student t distributed then the following properties apply:

Properties:

  1. These measurements are symmetrically distributed about the mean.
  2. A histogram of these measurements will be bell shaped.

Notes:

  1. The empirical rule does not apply. The Student t with mean m and standard deviation s has fatter tails than the corresponding Normal distribution with the same mean and standard deviation.
  2. The shape (i.e. fatness of the tails) is dependent on the degrees of freedom with fatter tails associated with smaller degrees of freedom.
  3. For df>=30, the Student t is (to all intents and purposes) indistinguishable from the normal distribution.
  4. For mean m=0 and standard deviation s=1 it is referred to as the Standard Student t distribution.
  5. As for the normal distribution, tables are available for Standard Student t distributions. However, since the shape is determined by the df, these tables are summarized into one table indexed by df (see the tables handed out in class).
  6. For mean m/=0 and standard deviation s/=1 we may use the transformation rule:
    t=(x - m)/ s

Table Tips:

  1. Scan the df column to find the row associated with the required df. The row represents the summary table for that Standard Student t.
  2. The desired t value (or its absolute value) is in the body of the table. The corresponding column heading is the proportion greater than t.
  3. If the desired t value is between two entries in the table then merely state the desired proportion as being between the proportions indicated by the column headings.

Work these problems for practice. Also, try these problems for additional practice.