Statistical Methods for Psychology

variety of settings. Suppose we hypothesize that when people know each other they tend to

be more accepting of individual differences. As a test of this hypothesis, we asked a group

of first-year male students matriculating at a small college to rate 12 target subjects (also

male) on physical appearance (higher scores represent greater attractiveness). At the end of

the first semester, when students have come to know one another, we again ask them to rate

those same 12 targets. Assume we obtain the data in Table 5.5, where each entry is the me-

dian rating that person (target) received when judged by participants in the experiment on a

30 point scale.

The gain score in this table was computed by subtracting the score obtained at the be-

ginning of the semester from the one obtained at the end of the semester. For example, the

first target was rated 3 points higher at the end of the semester than at the beginning. No-

tice that in 10 of the 12 cases the score at the end of the semester was higher than at the be-

ginning. In other words, the sign was positive. (The sign test gets its name from the fact

that we look at the sign, but not the magnitude, of the difference.)

Consider the null hypothesis in this example. If familiarity does not affect ratings of

physical appearance, we would not expect a systematic change in ratings (assuming that no

other variables are involved). Ignoring tied scores, which we don’t have anyway, we would

expect that by chance about half the ratings would increase and half the ratings would

decrease over the course of the semester. Thus, under , p(higher) 5 p(lower) 5 .50. The

binomial can now be used to compute the probability of obtaining at least 10 out of 12 im-

provements if is true:

From these calculations we see that the probability of at least 10 improvements 5 .0161 1

.0029 1 .0002 5 .0192 if the null hypothesis is true and ratings are unaffected by familiar-

ity. Because this probability is less than our traditional cutoff of .05, we will reject and

conclude that ratings of appearance have increased over the course of the semester. (Al-

though variables other than familiarity could explain this difference, at the very least our

test has shown that there is a significant difference to be explained.)

5.10 The Multinomial Distribution

The binomial distribution we have just examined is a special case of a more general distri-

bution, the multinomial distribution.In binomial distributions, we deal with events that

can have only one of two outcomes—a coin could land heads or tails, a wine could be

judged as more expensive or less expensive, and so on. In many situations, however, an

H 0

p(12)=

12!

12!0!

(.5)^12 (.5)^0 =.0002

p(11)=

12!

11!1!

(.5)^11 (.5)^1 =.0029

p(10)=

12!

10!2!

(.5)^10 (.5)^2 =.0161

H 0

H 0

Section 5.10 The Multinomial Distribution 133

Table 5.5 Median ratings of physical appearance at the beginning

and end of the semester

Target 12345 6 789101112

Beginning 12 21 10 8 14 18 25 7 16 13 20 15

End 15 22 16 14 17 16 24 8 19 14 28 18

Gain 3166322 2 11 3 1 8 3

multinomial
distribution