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

570 CHAPTER 17 | The Chi-Square Statistic: Tests for Goodness of Fit and Independence
■ Locating the Critical Region for a Chi-Square Test
Recall that a large value for the chi-square statistic indicates a big discrepancy between the
data and the hypothesis, and suggests that we reject H 0
. To determine whether a particular
chi-square value is significantly large, you must consult the table entitled The Chi-Square
Distribution (Appendix B). A portion of the chi-square table is shown in Table 17.2. The
first column lists df values for the chi-square test, and the top row of the table lists proportions
(alpha levels) in the extreme right-hand tail of the distribution. The numbers in
the body of the table are the critical values of chi-square. The table shows, for example,
that when the null hypothesis is true and df = 3, only 5% (.05) of the chi-square values
are greater than 7.81, and only 1% (.01) are greater than 11.34. Thus, with df = 3, any
chi-square value greater than 7.81 has a probability of p < .05, and any value greater
than 11.34 has a probability of p < .01.
TABLE 17.2
A portion of the table of
critical values for the
chi-square distribution.
df
Proportion in Critical Region
0.10 0.05 0.025 0.01 0.005
1 2.71 3.84 5.02 6.63 7.88
2 4.61 5.99 7.38 9.21 10.60
3 6.25 7.81 9.35 11.34 12.84
4 7.78 9.49 11.14 13.28 14.86
5 9.24 11.07 12.83 15.09 16.75
6 10.64 12.59 14.45 16.81 18.55
7 12.02 14.07 16.01 18.48 20.28
8 13.36 15.51 17.53 20.09 21.96
9 14.68 16.92 19.02 21.67 23.59
■ A Complete Chi-Square Test for Goodness of Fit
We use the same step-by-step process for testing hypotheses with chi-square as we
used for other hypothesis tests. In general, the steps consist of stating the hypotheses,
locating the critical region, computing the test statistic, and making a decision about H 0
.
The following example demonstrates the complete process of hypothesis testing with the
goodness-of-fit test.
EXAMPLE 17.2
A psychologist examining art appreciation selected an abstract painting that had no obvious
top or bottom. Hangers were placed on the painting so that it could be hung with
any one of the four sides at the top. The painting was shown to a sample of n = 50 participants,
and each was asked to hang the painting in the orientation that looked correct.
The following data indicate how many people chose each of the four sides to be placed
at the top:
Top Up
(Correct)
Bottom
Up
Left
Side Up
Right
Side Up
18 17 7 8
The question for the hypothesis test is whether there are any preferences among the four
possible orientations. Are any of the orientations selected more (or less) often than would
be expected simply by chance?