Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 9.1 | The t Statistic: An Alternative to z 273
For example, with df = 3, exactly 5% of the t distribution is located in the tail beyond
t = 2.353 (Figure 9.2). The process of finding this value is highlighted in Table 9.1. Begin
by locating df = 3 in the first column of the table. Then locate a proportion of 0.05 (5%) in
the one-tail proportion row. When you line up these two values in the table, you should find
t = 2.353. Because the distribution is symmetrical, 5% of the t distribution is also located
in the tail beyond t = –2.353 (see Figure 9.2). Finally, notice that a total of 10% (or 0.10) is
contained in the two tails beyond t = ±2.353 (check the proportion value in the “two-tails
combined” row at the top of the table).
TABLE 9.1
A portion of the t-distribution table. The numbers in the table are the values of t that separate the
tail from the main body of the distribution. Proportions for one or two tails are listed at the top of
the table, and df values for t are listed in the first column.
Proportion in One Tail
0.25 0.10 0.05 0.025 0.01 0.005
Proportion in Two Tails Combined
df 0.50 0.20 0.10 0.05 0.02 0.01
1 1.000 3.078 6.314 12.706 31.821 63.657
2 0.816 1.886 2.920 4.303 6.965 9.925
3 0.765 1.638 2.353 3.182 4.541 5.841
4 0.741 1.533 2.132 2.776 3.747 4.604
5 0.727 1.476 2.015 2.571 3.365 4.032
6 0.718 1.440 1.943 2.447 3.143 3.707
A close inspection of the t distribution table in Appendix B will demonstrate a point
we made earlier: as the value for df increases, the t distribution becomes more similar to a
normal distribution. For example, examine the column containing t values for a 0.05 proportion
in two tails. You will find that when df = 1, the t values that separate the extreme
5% (0.05) from the rest of the distribution are t = ±12.706. As you read down the column,
however, you should find that the critical t values become smaller and smaller, ultimately
reaching ±1.96. You should recognize ±1.96 as the z-score values that separate the
extreme 5% in a normal distribution. Thus, as df increases, the proportions in a t distribution
become more like the proportions in a normal distribution. When the sample size (and
degrees of freedom) is sufficiently large, the difference between a t distribution and the
normal distribution becomes negligible.
FIGURE 9.2
The t distribution with
df = 3. Note that 5% of the
distribution is located in the
tail beyond t = 2.353. Also,
5% is in the tail beyond
t = –2.353. Thus, a total
proportion of 10% (0.10) is
in the two tails combined.
5% 5%
22.353 0
2.353
t