240 Fundamentals of Statistics
You’ve looked at two approaches to statistical inference: the confi dence
interval and the hypothesis test. For a particular value of a, the width of the
confi dence interval around the sample average is equal to the width of the
two-sided acceptance region around m 0. This means that the following two
statements imply each other:- The value m 0 , lies outside the 11 2a 2 confi dence interval around x.
- Reject the null hypothesis that m 5 m 0 at the a signifi cance level.
The t Distribution
Up to now, you’ve been assuming that the value of s is known. What if
you didn’t know the value of s? One solution is to substitute the standard
deviation of the sample values, s for s in the hypothesis-testing equations.
However, there are problems with this approach. If s underestimates s, then
you’ll overestimate the signifi cance of the results, perhaps causing you to
reject the null hypothesis falsely. Or if s overestimates the value of s, you
could accept the null hypothesis when the null hypothesis isn’t true.
Early in the twentieth century, William Gosset, working at the Guinness
brewery in Ireland, became worried about the uncertainty caused by substi-
tuting s for s. He believed that the resulting error could be especially bad for
small sample sizes. What Gosset discovered was that when you substitute s
for s, the ratiox 2 m
s@"ndoes not follow the standard normal distribution; rather, it follows a distri-
bution called the t distribution.
The t distribution is a probability distribution centered around zero and
characterized by a single parameter called the degrees of freedom, which is
equal to the sample size, n, minus 1. For example, if the sample size is 20,
the degrees of freedom equal 19. The t distribution is similar to a standard
normal distribution except that it has heavier tails. As the sample size in-
creases, the t distribution approaches the standard normal, but for smaller
sample sizes there can be big differences.