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Chi-Square Tests

Chapter 12

12-2

Chapter Outline

12.1 Chi-Square Goodness of Fit Tests

12.2 A Chi-Square Test for Independence

12-3

12.1 Chi-Square Goodness of Fit Tests

1. Carry out n identical trials with k possible outcomes of each trial

2. Probabilities are denoted p1, p2, … , pk where p1 + p2 + … + pk = 1

3. The trials are independent4. The results are observed

frequencies, f1, f2, …, fk

12-4

Chi-Square Goodness of Fit Tests Continued

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

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

Test statistic:

Reject H0 if2 >

2 or p-value < 2 and the p-value are based on p-1

degrees of freedom

k

i i

ii

E

Ef=

1

22 )(

12-6

Example 12.1: The Microwave Oven Preference Case

Tables 12.1 and 12.2

12-7

Example 12.1: The Microwave Oven Preference Case Continued

7786.860

6057

120

120120

140

140121

80

80102

not true is hypothesis null stated previously the:

15.,30.,35.,20.:

2222

4

1

22

1

43210

k

i i

ii

E

Ef

H

H

12-8

Example 12.1: The Microwave Oven Preference Case #3

Figure 12.1

12-9

A Goodness of Fit Test for Multinomial Probabilities

fi = the number of items classified into group i

Ei = npi

H0: The values of the multinomial probabilities are p1, p2,…pk

H1: At least one of the multinomial probabilities is not equal to the value stated in H0

k

i i

ii

E

Ef

1

22

12-10

A Goodness of Fit Test for a 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

12-11

A Goodness of Fit Test for a Normal Distribution Continued

1. H0: the population has a normal distribution

2. Select random sample3. Define k intervals for the test4. Record observed frequencies5. Calculate the expected frequencies6. Calculate the chi-square statistics7. Make a decision

k

i i

oi

E

Ef

1

22

12-12

12.2 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 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

n

crE jiij ˆ

12-13

A Chi-Square Test for Independence Continued

H0: the two classifications are statistically independent

Ha: the two classifications are statistically dependent

Test statistic

Reject H0 if 2 > 2 or if p-value <

2 and the p-value are based on (r-1)(c-

1) degrees of freedom

cellsall

22

ˆ)ˆ(

ij

ijij

E

Ef=

12-14

Example 12.3 The Client Satisfaction Case

Table 12.4

12-15

Example 12.3 The Client Satisfaction Case #2

Figure 12.2 (a)

12-16

Example 12.3 The Client Satisfaction Case #3

H0: fund time and level of client satisfaction are independentH1: fund time and level of client satisfaction are dependent

Calculate frequencies under independence assumption

Calculate test statistic of 46.44Reject H0