Statistical Methods for Psychology

our hypothetical distribution of differences when the null hypothesis is false, and the alter-

native hypothesis ( ) is true. Remember that the distribution for is only hypothetical.

We really do not know the location of that distribution, other than that it is higher (greater

differences) than the distribution of. (I have arbitrarily drawn that distribution so that

its mean is 2 units above the mean under H 0 .)

The darkly shaded portion in the top half of Figure 4.3 represents the rejection region.

Any observation falling in that area (i.e., to the right of about 3.5) would lead to rejection

of the null hypothesis. If the null hypothesis is true, we know that our observation will fall

in this area 5% of the time. Thus, we will make a Type I error 5% of the time.

The cross hatched portion in the bottom half of Figure 4.3 represents the probability

(b) of a Type II error. This is the situation in which having someone waiting makes a dif-

ference in leaving time, but whose value is not sufficiently high to cause us to reject.

In the particular situation illustrated in Figure 4.3, we can in fact calculate bby using

the normal distribution to calculate the probability of obtaining a score less than3.5 (the

critical value) if m535 and s515 for each condition. The actual calculation is not im-

portant for your understanding of b; because this chapter was designed specifically to

avoid calculation, I will simply state that this probability (i.e., the area labeled b) is .76.

Thus for this example, 76% of the occasions when waiting times (in the population) differ

by 3.5 seconds (i.e., is actually true), we will make a Type II error by failing to reject

when it is false.

From Figure 4.3 you can see that if we were to reduce the level of a(the probability of

a Type I error) from .05 to .01 by moving the rejection region to the right, it would reduce

the probability of Type I errors but would increase the probability of Type II errors. Setting

aat .01 would mean that b5.92. Obviously there is room for debate over what level of

significance to use. The decision rests primarily on your opinion concerning the relative

importance of Type I and Type II errors for the kind of study you are conducting. If it were

H 0

H 1

H 0

H 0

H 1 H 1

98 Chapter 4 Sampling Distributions and Hypothesis Testing

0.4

0.2

0.0

–6 –4 –2 0

Difference in means

H 0 = True

246

y

0.4

0.2

0.0

–6 –4 –2 0

Difference in means

H 0 = False

24

H 0 H 1

6

y

Figure 4.3 Distribution of mean differences under H 0 and H 1