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

SECTION 17.1 | Introduction to Chi-Square: The Test for Goodness of Fit 565
total sample size: Σ f o
= n. Finally, you should realize that we are not assigning individuals
to categories. Instead, we are simply measuring individuals to determine the category in
which they belong.
DEFINITION
The observed frequency is the number of individuals from the sample who are
classified in a particular category. Each individual is counted in one and only one
category.
■ Expected Frequencies
The general goal of the chi-square test for goodness of fit is to compare the data (the
observed frequencies) with the null hypothesis. The problem is to determine how well the
data fit the distribution specified in H 0
—hence the name goodness of fit.
The first step in the chi-square test is to construct a hypothetical sample that represents
how the sample distribution would look if it were in perfect agreement with the proportions
stated in the null hypothesis. Suppose, for example, the null hypothesis states that the
population is distributed in three categories with the following proportions:
Category A Category B Category C
H 0
: 25% 50% 25%
(The population is distributed across
the three categories with 25% in
category A, 50% in category B,
and 25% in category C.)
If this hypothesis is correct, how would you expect a random sample of n = 40 individuals
to be distributed among the three categories? It should be clear that your best strategy
is to predict that 25% of the sample would be in category A, 50% would be in category B,
and 25% would be in category C. To find the exact frequency expected for each category,
multiply the sample size (n) by the proportion (or percentage) from the null hypothesis. For
this example, you would expect
25% of 40 = 0.25(40) = 10 individuals in category A
50% of 40 = 0.50(40) = 20 individuals in category B
25% of 40 = 0.25(40) = 10 individuals in category C
The frequency values predicted from the null hypothesis are called expected frequencies.
The symbol for expected frequency is f e
, and the expected frequency for each category is
computed by
expected frequency = f e
= pn (17.1)
where p is the proportion stated in the null hypothesis and n is the sample size.
DEFINITION
The expected frequency for each category is the frequency value that is predicted
from the proportions in the null hypothesis and the sample size (n). The expected
frequencies define an ideal, hypothetical sample distribution that would be obtained
if the sample proportions were in perfect agreement with the proportions specified
in the null hypothesis.