Statistical Methods for Psychology

important to avoid Type I errors (such as falsely claiming that the average driver is rude),

then you would set a stringent (i.e., small) level of a. If, on the other hand, you want to

avoid Type II errors (patting everyone on the head for being polite when actually they are

not), you might set a fairly high level of a. (Setting a5.20 in this example would reduce

bto .46.) Unfortunately, in practice most people choose an arbitrary level of a, such as .05

or .01, and simply ignore b. In many cases this may be all you can do. (In fact you will

probably use the alpha level that your instructor recommends.) In other cases, however,

there is much more you can do, as you will see in Chapter 8.

I should stress again that Figure 4.3 is purely hypothetical. I was able to draw the fig-

ure only because I arbitrarily decided that the population means differed by 2 units, and the

standard deviation of each population was 15. The answers would be different if I had cho-

sen to draw it with a difference of 2.5 and/or a standard deviation of 10. In most everyday

situations we do not know the mean and the variance of that distribution and can make only

educated guesses, thus providing only crude estimates of b. In practice we can select a

value of munder that represents the minimumdifference we would like to be able to de-

tect, since larger differences will have even smaller bs.

From this discussion of Type I and Type II errors we can summarize the decision-

making process with a simple table. Table 4.1 presents the four possible outcomes of an

experiment. The items in this table should be self-explanatory, but there is one concept—

power—that we have not yet discussed. The powerof a test is the probability of rejecting

when it is actually false. Because the probability of failingto reject a false is b, then

power must equal 12 b.Those who want to know more about power and its calculation

will find power covered in Chapter 8.

4.8 One- and Two-Tailed Tests

The preceding discussion brings us to a consideration of one- and two-tailed tests. In our

parking lot example we were concerned if people took longer when there was someone

waiting, and we decided to reject only if a those drivers took longer. In fact, I chose that

approach simply to make the example clearer. However, suppose our drivers left 16.88 sec-

onds soonerwhen someone was waiting. Although this is an extremely unlikely event to

observe if the null hypothesis is true, it would not fall in the rejection region, which con-

sisted solelyof long times. As a result we find ourselves in the position of not rejecting

in the face of a piece of data that is very unlikely, but not in the direction expected.

The question then arises as to how we can protect ourselves against this type of situa-

tion (if protection is thought necessary). One answer is to specify before we run the experi-

ment that we are going to reject a given percentage (say 5%) of the extremeoutcomes, both

those that are extremely high and those that are extremely low. But if we reject the lowest

5% and the highest 5%, then we would in fact reject H 0 a total of 10% of the time when it

H 0

H 0

H 0 H 0

H 1

Section 4.8 One- and Two-Tailed Tests 99

Table 4.1 Possible outcomes of the decision-making process

True State of the World

Decision H 0 True H 0 False

RejectH 0 Type I error p5a Correct decision p 5 1 – b5Power

Don’t rejectH 0 Correct decision p 5 1 – a Type II error p5b

power