1. Chapter 12
Chi-Square Tests

2. Chi-Square Tests 12.1 Chi-Square Goodness of Fit Tests
12.2 A Chi-Square Test for Independence

3. The Multinomial Experiment
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

4. 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)

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 if
2 > 2 or p-value < 
2 and the p-value are based on p-1 degrees of freedom

6. Example 12.1: The Microwave Oven Preference Case
H0: p1 = .20, p2 = .35, p3 = .30, p4 = .15
Ha: H0 fails to hold

7. Example 12.1: The Microwave Oven Preference Case Continued

8. 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

9. Example 12.2: The Car Mileage Case
Consider the 50 gas mileage samples from Chapter 1 (Table 1.4)
A histogram is symmetrical and bell-shaped
This suggests a normal distribution
Will test this using a chi-square goodness of fit test

10. Example 12.2: The Car Mileage Case #2
First divide the number line into intervals
Will use the class boundaries of the histogram in Figure 2.10
Table below gives intervals and observed frequencies

11. Example 12.2: The Car Mileage Case #3
Next step is to calculate expected frequencies
Those calculations are shown below

12. Example 12.2: The Car Mileage Case #4
Will test:
H0: Population is normally distributed
Ha: Population is not normally distributed
2= < 20.05= cannot reject H0

13. 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

14. 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

15. Example 12.3: The Client Satisfaction Case
A financial institution sells investment products
Examining whether customer satisfaction depend on the type of product purchased
Data shown in Table 12.4

16. Example 12.3: The MegaStat Output

17. Example 12.3: The MINITAB Output

18. Example 12.3: The Client Satisfaction Case
H0: Client satisfaction is independent of fund type
Ha: Client satisfaction depends upon fund type

19. Example 12.3: Plots of Row Percentages Versus Investment Type