Chi-Square Test
Section 12.1
Categorical Variables

Based on observations

Univariate – single categorical variable
 Example:

Sample 100 people & ask if they
agree or disagree with a question.
Bivariate – uses two categorical variables
 Example:
Sample 100 people & ask if they
are male/female and what political
party they support.
One-Way Frequency Table - univariate
Data
Democrat
Democrat
Democrat
Independent
Republican
Democrat
Republican
Independent
Republican
Republican
Republican
Republican
Vertical OneWay Table
Horizontal One-Way Table
Freq.
Democrat
Freq.
4
Republican
6
Independent
2
Democrat
4
Republican
6
Independent
2
Goodness of Fit Test

2
Used to measure the extent to which the
observed counts differ from the expected
counts.
 K = # categories of a catagorical variable
 Df = k – 1
2
Observed  Expected 

2
 Test Statistic:   
Expected

Assumptions

Observed Values are based on random
Samples

Sample size is large – each cell count is at
least 5.
Hypotheses

Ho: State each proportion’s hypothesized
value.

HA: At least 1 of the proportions differ
from the hypothesized value.
It uses the Chi-Square Chart
Positively Skewed
 Uses d.f.
 On calculator!

Is there a preference in type of car?
P1=proportion who prefer a SUV
Freq.
SUV
Expected
27
Truck
25
Sedan
29
Sports
19
P2=proportion who prefer a truck
p3=proportion who prefer a sedan
P4=proportion who prefer a sports car
 H o : p1  p2  p3  p4

 H A : at least 1 prop. is different
2 
Assumptions: Random Samples
& all cell counts are at least 5.
Use a Chi-Square goodness of
fit Test
 Observed - Predicted 
Predicted
 27  25
2 
25
 2  2.24
2
 25  25

25
2
 29  25

P-val = xcdf(2.24,∞, 3)=0.52
25
2
19  25 

25
2
A researcher believes that the number of homicides crimes in CA by
season is uniformly distributed. To test this claim, you randomly select
1200 homicides from a recent year and record the season when each
happened.
Season
Freq
Spring
312
Summer
299
Fall
297
Winter
293
Results from a previous survey asking people who go to movies at least
once a month are shown in the table below. To determine whether this
distribution is still the same, you randomly select 1000 people who go to
movies at least once a month and record the age of each. Are the
distributions the same?
Age
Survey
Freq
2 - 17
26.70%
240
18 - 24
19.80%
214
25 - 39
19.70%
183
40 - 49
14%
156
50+
19.80%
207
Homework

Worksheet