Introduction to Statistics
Steps in Significance/Hypothesis Testing
Using the Chi-Square (χ
χ2) Distribution
Step 1: State null (Ho) and alternative (Ha) hypotheses.
When we examine a relationship from sample data presented in a two-way table, we
want to know if the two variables are independent (no significant relationship) or related
in the population. Does the relationship we observe exist by chance or not?
When using the Chi-square (χ2) distribution to establish if a relationship exists between
two categorical variables, the null hypothesis (Ho) statement does not contain
population symbols. Rather, we make a written statement. For example, about social
class and formal education, we can write:
Ho: Education is not related to social class (education and social class are independent)
The alternative hypothesis (Ha) is also a written statement in which we declare a
relationship to exist between two variables. For example:
Ha: Education is related to social class.
It is important to note that hypotheses tested using the Chi-square distribution are not
two-tailed. χ2 values are always positive because the Chi-square distribution ranges from
zero to positive infinity.
Step 2: Compute a test statistic.
The test statistic is calculated using sample data in a two-way table. You use two
formulas: the first formula is for finding expected frequencies/counts (fe). In the second
formula, the χ2 test statistic, you compare the expected frequency/count (fe) to the
frequency/count you observe (fo) in the two-way table.
You must calculate fe for each cell in a two-way table:
fe =
[(RowM arg inal )x(ColumnM arg inal )]
TotalSampleSize
Then, for each cell in a two-way table, perform the chi-square calculation in [ ] below,
adding up the calculation for each cell.
 ( f − f e )2 
χ2 = ∑ o

fe


M. Berntson – Grinnell College
Spring 2002 – Steps in Hypothesis Testing using the Chi-Square Distribution
1
Step 3: Make a decision about the null hypothesis (Ho).
In this step, you will either reject or not reject Ho. When you reject Ho, you conclude in
favor of the alternative hypothesis (Ha) of a statistically significant relationship. When you
do not reject Ho, you conclude in favor of the null hypothesis of no statistically significant
relationship. There are two decision rules you may follow. You should know how to use
each one.
Decision Rule 1 instructions: Find a p-value and compare it to your significance level
(α).
P-values are the probabilities associated with the test statistic (observed χ2) found in
step 2 and degrees of freedom. Degrees of freedom [(#rows - 1)(#columns - 1)] and
probabilities organize the values of χ2 found in the body of Moore's Table E.
Use Moore's Table E and the instructions below to find the p-value. Note that when
finding a p-value in Table E, your observed χ2 might fall between two values of χ2. This
means that your p-value will fall between two probabilities.
Finding P-Values using Table E
Test Statistic
(observed χ2)
Look for χ2 in body of
table
P-value
given df [(r-1)(c-1)], find
probability in table
(this might be a range)
Decision rule 1: If range of p-values < α, reject Ho (eg: .01 to .025 < .05). If range of pvalues > α, do not reject Ho (eg: .15 to .20 > .05).
Decision Rule 2 instructions: Find a critical value (χ2*) and compare it to your test
statistic.
Critical values (χ2*) are the χ2 scores in Table E associated with your level of
significance (α) and degrees of freedom.
Finding Critical Values (χ2*) Using Table E
Critical value (χ2*)
given df [(r-1)(c-1)] and α,
find value of χ2 in table
Decision rule 2: If test statistic > χ2*, reject Ho. If test statistic < χ2*, do not reject Ho.
M. Berntson – Grinnell College
Spring 2002 – Steps in Hypothesis Testing using the Chi-Square Distribution
2