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Part III: Distributions and the Central Limit Theorem
you would have to look for 0.05 (rather than 10%, as you might be inclined to
do.) But using the bottom row of the table, you just look for 90%. (The result
you get using either method ends up being in the same column.)
When looking for t*-values for confidence intervals, use the bottom row of the

t-table as your guide, rather than the headings at the top of the table.
Studying Behavior Using the t-Table
You can use computer software to calculate any probabilities, percentiles, or
critical values you need for any t-distribution (or any other distribution) if it’s
available to you. (On exams it may not be available.) However, one of the nice
things about using a table to find probabilities (rather than using computer
software) is that the table can tell you information about the behavior of the
distribution itself — that is, it can give you the big picture. Here are some
nuggets of big-picture information about the t-distribution you can glean by
scanning the t-table (in the appendix).
In Figure 10-2, as the degrees of freedom increase, the values on each
t-distribution become more concentrated around the mean, eventually resem-
bling the Z-distribution. The t-table confirms this pattern as well. Because
of the way the t-table is set up, if you choose any column and move down
through the numbers in the column, you’re increasing the degrees of freedom
(and sample size) and keeping the right-tail probability the same. As you do
this, you see the t-values getting smaller and smaller, indicating the t-values
are becoming closer to (hence more concentrated around) the mean.
I labeled the second-to-last row of the t-table with a z in the df column. This
indicates the “limit” of the t-values as the sample size (n) goes to infinity. The
t-values in this row are approximately the same as the z-values on the Z-table
(in the appendix) that correspond to the same greater-than probabilities. This
confirms what you already know: As the sample size increases, the t- and the
Z-distributions look more and more alike. For example, the t-value in row 30
of the t-table corresponding to a right-tail probability of 0.05 (column 0.05) is
1.697. This lies close to z = 1.645, the value corresponding to a right-tail area of
0.05 on the Z-distribution. (See row Z of the t-table.)
It doesn’t take a super-large sample size for the values on the t-distribution
to get close to the values on a Z-distribution. For example, when n = 31 and
df = 30, the values in the t-table are already quite close to the corresponding
values on the Z-table.

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