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 561
17.1 Introduction to Chi-Square: The Test for Goodness of Fit
LEARNING OBJECTIVES
1. Describe parametric and nonparametric hypothesis tests.
2. Describe the data (observed frequencies) for a chi-square test for goodness of fit.
3. Describe the hypotheses for a chi-square test for goodness of fit and explain how the
expected frequencies are obtained.
■ Parametric and Nonparametric Statistical Tests
All the statistical tests we have examined thus far are designed to test hypotheses about
specific population parameters. For example, we used t tests to assess hypotheses about
a population mean (μ) or mean difference (μ 1
– μ 2
). In addition, these tests typically
make assumptions about other population parameters. Recall that, for analysis of variance
(ANOVA), the population distributions are assumed to be normal and homogeneity of
variance is required. Because these tests all concern parameters and require assumptions
about parameters, they are called parametric tests.
Another general characteristic of parametric tests is that they require a numerical score
for each individual in the sample. The scores then are added, squared, averaged, and otherwise
manipulated using basic arithmetic. In terms of measurement scales, parametric tests
require data from an interval or a ratio scale (see Chapter 1).
Often, researchers are confronted with experimental situations that do not conform to
the requirements of parametric tests. In these situations, it may not be appropriate to use a
parametric test. Remember that when the assumptions of a test are violated, the test may
lead to an erroneous interpretation of the data. Fortunately, there are several hypothesistesting
techniques that provide alternatives to parametric tests. These alternatives are called
nonparametric tests.
In this chapter, we introduce two commonly used examples of nonparametric tests. Both
tests are based on a statistic known as chi-square and both tests use sample data to evaluate
hypotheses about the proportions or relationships that exist within populations. Note that
the two chi-square tests, like most nonparametric tests, do not state hypotheses in terms of
a specific parameter and they make few (if any) assumptions about the population distribution.
For the latter reason, nonparametric tests sometimes are called distribution-free tests.
One of the most obvious differences between parametric and nonparametric tests is
the type of data they use. All of the parametric tests that we have examined so far require
numerical scores. For nonparametric tests, on the other hand, the participants are usually
just classified into categories such as Democrat and Republican, or High, Medium, and
Low IQ. Note that these classifications involve measurement on nominal or ordinal scales,
and they do not produce numerical values that can be used to calculate means and variances.
Instead, the data for many nonparametric tests are simply frequencies—for example,
the number of Democrats and the number of Republicans in a sample of n = 100 registered
voters.
Occasionally, you have a choice between using a parametric and a nonparametric test.
Changing to a nonparametric test usually involves transforming the data from numerical
scores to nonnumerical categories. For example, you could start with numerical scores
measuring self-esteem and create three categories consisting of high, medium, and low
self-esteem. In most situations, the parametric test is preferred because it is more likely