Chi-Square Tests
• Categorical data
• 1-sample, compared to theoretical distribution
– Goodness-of-Fit Test
• 2+ samples, 2+ levels of response variable
–
Chi-square Test
Chi-Square Tests Slide #1

Goodness-of-Fit Test
• Compare observed to theoretical frequencies of individuals in categories.
• Examples –
– Test whether responses are “random” (e.g., preference)
– Test Mendelian genetics (e.g., 3:1 and 9:3:3:1 theories).
– Test use of available resources (e.g., compare habitat usage to availability).
Chi-Square Tests Slide #2

An Illustrative Example
• Determine, at the 10% level, if Northland students prefer the Chris Duarte Group (CDG), Ronnie
Baker Brooks (RBB), or Bernard Allison (BA).
• Hypotheses?
• H a
: “different # of students prefer each artist”
• H o
: “same # of students prefer each artist”
Chi-Square Tests Slide #3

An Illustrative Example
• Under H o
, what proportion prefer each artist?
1/3
• If n=78, how many students prefer each artist if H o is true?
26
Expected
Table
Artist CDG
Freq 26
RBB
26
BA
26
Chi-Square Tests Slide #4

An Illustrative Example
• Suppose these results were obtained:
Observed
Table
Artist CDG
Freq 24
RBB
38
BA
16
• Is there a preference – i.e., are these observations significantly different from what was expected when assuming no preference?
Chi-Square Tests Slide #5

A New Test Statistic
 2   table
 observed exp
 exp ected ected

2 df = cells - 1
Chi-Square Tests Slide #6

An Illustrative Example
Observed
Table
Artist CDG
# 24
RBB
38
BA
16
Expected
Table
Artist CDG
# 26
RBB

2 =

24

26

2
26


38

26

2
26


2 = 0.15 + 5.54 + 3.85 = 9.54

16

26

2
26
26 df = (3-1) = 2
Conclusion?
p-value = 0.00848
BA
26
Chi-Square Tests Slide #7

Goodness-of-Fit Test
•
H o
: distribution of individuals into levels follows the theoretical distribution
•
H
A
: distribution of individuals into levels does
NOT follow the theoretical distribution
•
Sample: randomized, single variable of size n
•
Assume: at least 5 in each cell of expected table
•
Statistic: Observed frequency table
Chi-Square Tests Slide #8

Goodness-of-Fit Test
•
Test Statistic:  2   table
 observed exp
 exp ected ected

2
• df: cells-1
•
Confidence Region: pˆ
 z
* pˆ

1
 pˆ
 n
–
Chi-Square Tests Slide #9

Examine HO – Page 5
Chi-Square Tests Slide #10