The document describes the chi-square test, a non-parametric test used to determine if distributions of categorical variables differ from expected distributions. It can test differences between observed and expected data distributions for one sample or multiple independent samples. The chi-square test involves defining hypotheses, determining degrees of freedom, calculating the test statistic, finding the p-value, and deciding whether to reject the null hypothesis based on a significance level. If the null hypothesis is rejected, the variables are considered not independent or homogeneous.

1. CHI-SQUARE TEST X2
Another most widely used test of
significance ( non- parametric).
Particularly useful in test involving
cases where persons, events or
objects are grouped in two or more
nominal categories such as yes or no,
approved-undecided disapprove or
class “A,B,C,D.”

2. • By using the Chi-square , we can test for
the significant differences between the
observed distribution of data among
categories and expected distribution of
data based upon the null hypothesis.
• It is useful in case of one-sample
analysis, two independent samples or k
independent samples

3. 1. One variable Chi-square( goodness- of –fit test)
Compares a set of observed frequency (0) for each categories.
( Example: political candidates, contestant 1,2,3)
2. Two variable chi- square( test of independence)
With two or more categories/ produce to determine whether
two or more variables are statistically independent
( Example: Classification of heights as first variable and weight as
second variable)
3. Test of Homogeneity the test is concerned with two or more
samples, with only one criterion variable.
 It is used to determine if two or more populations are
homogeneous.
X2

4. • 1.Make a problem statement
• 2. Hypotheses
• H0: The distributions of the two
populations are the same.
•
Ha: The distributions of the two
populations are not the same.

5. • The significance level, α corresponds to
the size of the rejection region.
It determines how small the p-value
should be in order to reject the null
hypothesis.
The common choices for α are 0.05, 0.01,
or 0.10.

6. P-value
• The p-value is the probability of getting
a value for the test statistic as large or
larger than the observed value of the
test statistic just by random chance.
• To determine a p-value look at the χ2
table with df degrees of freedom and
find where the observed value of the χ2
statistic falls on this table.

7. Level of Significance:
• Degrees of freedom.
• DF = (r - 1) (c - 1)
• where r is the number of
populations/no. of rows, and
• c is the number of levels for the
categorical variable/no. of
columns.

8. • X2 = ∑
• Where:
• X2 = the chi-square test
• O = the observed frequencies
• E = the expected frequencies

9. • there are two ways to make a decision in
this test
• Classical
Reject null hypothesis if χ2 ≥ χ2
α,df
Fail to reject null hypothesis if χ2 < χ2
α,df
OR
P-value:
• this method is preferred by researchers
currently conducting research
Reject null hypothesis if p-value ≤ α
Fail to reject null hypothesis if p-value > α

10. • The conclusion is a statement written to
convey the results of the research. If possible,
avoid statistical terminology and should be
written in a form that can be easily understood
by non-statisticians.
• Example
• If the null hypothesis is rejected then conclude that
the two variables are not homogeneous at the
specified significance level.
• If the null hypothesis is not rejected then conclude
that the variables are homogeneous at the specified
significance level.