Statistical Analysis for Education and Psychology Researchers

where n is the number of matched pairs (ties, if present, would not be included and n
reduced accordingly), s+ is the number of designated + signs (count of
observations>median) and f is the count of − signs. If the square of the difference
between the number of + and − signs is ≥ four times the sample size then the null
hypothesis of no difference is rejected.

Interpretation

In this example, s+=5; f=7; and n=12. Since (s+−f)^2 =(5−7)^2 =4, which is <48, (4×12) the
null hypothesis is not rejected and it is concluded that the average performance of
students under the two conditions is not significantly different. If an investigator wanted
to use alternative alpha levels the following comparable formula could be used:
Alternative hypothesis Alpha Formula
two-tailed 1% (s−f)^2 ≥6.7n
one-tailed 5% (s−f)^2 ≥2.7n
one-tailed 1% (s−f)^2 ≥5.4n

This simple test is particularly useful for detecting significant change in pre-test/ post-test
designs.

6.6 r×k Chi-square Test

When to Use

This test is an extension of either the r×2 Sample χ^2 test of homogeneity when there are
more than two independent groups (samples), or the one sample χ^2 test of independence
when there are more than two categories in the row or column variables. In the
homogeneity design k independent random samples are drawn (the columns’ variable)
from each of k populations and the distribution of proportions in the r (row variable)
categories are compared hence the term r×k Chi-square. An alternative model is the test
of independence when a single random sample is drawn from a single population of
subjects with two categorical measures for each subject, that is the row and column
variables. For the homogeneity design an investigator would be interested in the effect of
the column variable on the response or row variable. For the independent design an
investigator would be interested in testing whether the two categorical variables are
independent.

Statistical Inference and Null Hypothesis

For the homogeneity design the general form of the null hypothesis is that the k
independent samples come from one single population (or identical populations). This
may be stated as the proportions of subjects in each category of the row variable
(measured variable) is the same in each of the k independent groups (samples) of the
column variable. The parameters being estimated are the proportions in each cell of the
contingency table which the observed sample proportions are intended to estimate, The

Statistical analysis for education and psychology researchers 192