Statistical Methods for Psychology

and substitute sfor sto give

Since we know that for any particular sample, is more likely than not to be smaller than

the appropriate value of , we can see that the t formula is more likely than not to produce

a larger answer (in absolute terms) than we would have obtained if we had solved for z us-

ing the true but unknown value of itself. (You can see this in Figure 7.4, where more

than half of the observations fall to the left of .) As a result, it would not be fair to treat

the answer as a zscore and use the table of z. To do so would give us too many “signifi-

cant” results—that is, we would make more than 5% Type I errors. (For example, when we

were calculating z, we rejected at the .05 level of significance whenever zexceeded

6 1.96. If we create a situation in which is true, repeatedly draw samples of n 5 5, and

use in place of , we will obtain a value of 6 1.96 or greater more than 10% of the time.

The cutoff in this case is 2.776.)

The solution to our problem was supplied in 1908 by William Gosset, who worked for

the Guinness Brewing Company, published under the pseudonym of Student, and wrote

several extremely important papers in the early 1900s. Gosset showed that if the data are

sampled from a normal distribution, using in place of would lead to a particular sam-

pling distribution, now generally known as Student’s tdistribution.As a result of

Gosset’s work, all we have to do is substitute , denote the answer as t, and evaluate t with

respect to its own distribution, much as we evaluated zwith respect to the normal distribu-

tion. The t distribution is tabled in Appendix t, and examples of the actual distribution of t

for various sample sizes are shown graphically in Figure 7.5.

As you can see from Figure 7.5, the distribution of t varies as a function of the degrees

of freedom, which for the moment we will define as one less than the number of observations

s^2

s^2 s^2

t.05

s^2 s^2

H 0

H 0

s^2

s^2

s^2

s^2

t=

X2m

sX =

X2m

s

2 n

=

X2m

B

s^2

n

186 Chapter 7 Hypothesis Tests Applied to Means

Figure 7.4 Sampling distribution of the sample variance

Sample variance

0.020.040.060.080.0100.0120.0140.0160.0180.0200.0220.0240.0260.0280.0300.0320.0

8000

6000

4000

2000

0

Std. Dev = 35.04

Mean = 49.9

N = 50000.00

Student’s t
distribution