Statistical Analysis for Education and Psychology Researchers

Sampling distribution of thestatistic χ^2 stastistic

If you are unfamiliar with hypothesis testing you might like to come back to this section
after having read ‘Hypothesis testing’ (pp. 108–113). Consider the sampling distribution
of the χ^2 statistic. This statistic is often used to answer questions of the type, ‘Is there a
relationship between two categorical variables such as school type and identity status?’
For example, we may select a random sample of pupils and cross classify them on two
variables to see whether there is a statistically significant association between school
type, (one variable, two categories private or state), and the other variable, identity status,
four categories;—achievement, moratorium, foreclosure and diffusion. In another type of
design the χ^2 statistic can be used to compare the distribution of proportions in two
populations. When there are two separate and independent random samples (sample size
fixed by the research design) drawn from two populations, for example, boys and girls or
state schools and private schools, a χ^2 test of homogeneity would be appropriate. Use of
the χ^2 statistic in various research designs is discussed in Chapter 6.
A research question addressed by investigators might be: ‘Is there a relationship
between school type and identity status’ (See for example, an empirical study by Roker
and Banks, 1993). The corresponding null hypothesis would be that the population
proportions in the four identity states is equal in the two populations of students
(private/state school). The Null Hypothesis is a hypothesis of no difference and plays a
crucial role in statistical analysis (it is sometimes called the statistical hypothesis). This is
the same as saying the distribution of the population proportions, the parameter pi, π, is
the same in each population. This is illustrated in Table 4.1.

Table 4.1: Table of parameter distribution for a

two-way table

(^) Population (columns)
Outcome (rows) 1 (Private) 2 (State)
Achievement 1 π1(1) π1(2)
Moratorium 2 π2(1) π2(2)
Foreclosure 3 π3(1) π3(2)
Diffusion 4 π4(1) π4(2)
Σ 1 1
Looking at Table 4.1, another way of stating the null hypothesis, H 0 , is to say that the set
of parameters in column 1 is the same as the set of parameters in column 2. In notational
form:
H 0 :π1(1)=π1(2)
π2(1)=π2(2)
π3(1)=π3(2)
π4(1)=π4(2)
You could obtain empirically the sampling distribution of the χ^2 test statistic when H 0 is
true by drawing a very large number of pairs of random samples, a random sample for
Statistical analysis for education and psychology researchers 92