Download Chi-Square Tests - McGraw Hill Higher Education

Chapter 12
Chi-Square Tests
McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
Chi-Square Tests
12.1 Chi-Square Goodness of Fit Tests
12.2 A Chi-Square Test for Independence
12-2
LO 1: Test hypotheses
about multinomial
probabilities by using a
chi-square goodness of
fit test.



12.1 Chi-Square
Goodness of Fit Tests
Collect count data to study how counts are
distributed among cells
Often use categorical data for statistical
inference
May use a multinomial experiment

Similar to a binomial experiment only more than
two outcomes are possible
12-3
LO1
The Multinomial Experiment
1.
2.
3.
4.
Carry out n identical trials with k possible
outcomes of each trial
Probabilities are denoted p1, p2, … , pk
where p1 + p2 + … + pk = 1
The trials are independent
The results are observed frequencies of the
number of trials that result in each of k
possible outcomes, denoted f1, f2, …, fk
12-4
LO1



Chi-Square Goodness of Fit
Tests
Consider the outcome of a multinomial
experiment where each of n randomly
selected items is classified into one of k
groups
Let fi = number of items classified into group i
(ith observed frequency)
Ei = npi = expected number in ith group if pi is
probability of being in group i (ith expected
frequency)
12-5
LO1


A Goodness of Fit Test for
Multinomial Probabilities
H0: multinomial probabilities are p1, p2, … , pk
Ha: at least one of the probabilities differs from p1,
p2, … , pk
2
(
f

E
)
2
i
 = i
Ei
i 1
k

Test statistic:

Reject H0 if
 2 > 2 or p-value < 

2 and the p-value are based on p-1
degrees of freedom
12-6
LO 2: Perform a
goodness of fit test for
normality.
Normal Distribution



Have seen many statistical methods based
on the assumption of a normal distribution
Can check the validity of this assumption
using frequency distributions, stem-and-leaf
displays, histograms, and normal plots
Another approach is to use a chi-square
goodness of fit test
12-7
LO2
1.
2.
3.
4.
5.
6.
7.
A Goodness of Fit Test for a
Normal Distribution
Test the following null and alternative hypotheses:
H0: the population has a normal distribution
Ha: population does not have normal distribution
Select random sample and compute sample mean
and standard deviation
Define k intervals for the test
Record observed frequency (fi) for each interval
2
Calculate expected frequency (Ei)
k


f

E
2
i
i



Calculate the chi-square statistic
Ei
i 1
Make a decision
12-8
LO 3: Decide whether
two qualitative variables
are independent by
using a chi-square test
for independence.

Each of n randomly selected items is classified on
two dimensions into a contingency table with r rows
an c columns and let



12.2 A Chi-Square Test for
Independence
fij = observed cell frequency for ith row and jth column
ri = ith row total
cj = jth column total
Expected cell frequency for ith row and jth column
under independence
Eˆ ij 
ri c j
n
12-9
LO3
A Chi-Square Test for
Independence
Continued



H0: the two classifications are statistically
independent
Ha: the two classifications are statistically dependent
2
Test statistic
ˆ
(f E )
=
2
all cells


ij
ij
Eˆ ij
Reject H0 if 2 > 2 or if p-value < 
2 and the p-value are based on (r-1)(c-1) degrees
of freedom
12-10