1. Chapter 18 Chi-Square Tests

2. 2 Distribution
Let x1, x2, .. xn be a random sample from a normal distribution with  and 2, and let s2 be the sample variance, then the random variable (n-1)s2/2 has 2 distribution with n-1 degrees of freedom.
Probability Density Function, with k degrees of freedom,
Mean
Variance
Mode = k-2 (when k 3)

3. fr.academic.ru/pictures/frwiki/67/Chi-square_..
2 Distribution
fr.academic.ru/pictures/frwiki/67/Chi-square_..

4. 2 Distribution Goodness-of-fit Tests Tests of Independence
Tests of Homogeneity

5. Multinominal Experiments
A Multinomial experiment is a statistical experiment that has the following properties:
It consists of n repeated trials (repetitions).
Each trial can result in one of k possible outcomes.
The trials are independent.
The probabilities of the various events remain constant for each trial.

6. Goodness-of-fit Test
Observed Frequencies (Oi): Frequencies obtained from the actual performance of an experiment.
Goodness-of-fit Test: Test of null hypothesis that the observed frequencies follow certain pattern or theoretical distributions, expressed by the Expected Frequencies (Ei).

7. Goodness-of-fit Test for Multinominal Experiments
Degree of freedom = k -1, where k is the number of categories
Chi-square goodness-of-fit test is always a right-tailed test
Sample size should be large enough so that the expected frequency for each category is at least 5.

8. Goodness-of-fit Test for Multinominal Experiments
Null Hypothesis: H0: the observed frequencies follow certain pattern
Test statistic:
Degree of Freedom = k -1
Alt. Hypothesis
P-value
Rejection Criterion H1
P(2>02)
02 > 2,k-1

9. Goodness-of-fit Test for Multinominal Experiments -- Example 18.1
Department stores in shopping mall
H0: p1 = p2 = p3 = p4 = p5 = .20
H1: At least 2 of pi  .20
 = .01, df = 5 -1 = 4
Test statistic:
Critical value: .01,4 =
P-value =
Reject H0 Oi Ei
Oi-Ei
(Oi-Ei)2
(Oi-Ei)2/Ei 1
214
200 14
196
0.98 2
231 31
961
4.81 3
182
-18
324
1.62 4
219 19
361
1.81 5
154
-46
2116
10.58
19.79

10. Goodness-of-fit Test for Multinominal Experiments -- Example 18.2
Market shares
H0: p1=.144, p2=.181, p3=.248, p4=.141, p5=.149, p6 =.137
H1: At least 2 of pi are different
 = .025, df = 6 -1 = 5
Test statistic:
Critical value: .025,5 =
P-value = .1252
Fail to reject H0 Oi Ei
Oi-Ei
(Oi-Ei)2
(Oi-Ei)2/Ei 1
270
288
-18
324
1.1250 2
382
362 20
400
1.1050 3
467
496
-29
841
1.6956 4
317
282 35
1225
4.3440 5
298
-10
100
0.3356 6
276
274
0.0146
2000
8.6197

11. Test of Independence for a Contingency Table
Columns 1 2 … c
Rows
O11
O12
O1c u1
O21
O22
O2c u2 r
Or1
Or2
Orc ur 1 2 c

12. Test of Independence for a Contingency Table
Null Hypothesis: H0: The two attributes are independent
Alt. Hypothesis: H1: The two attributes are dependent
Test statistic:
Degree of Freedom = (r-1)(c-1)
Alt. Hypothesis
P-value
Rejection Criterion H1
P(2>02)
02 > 2,df

13. Test of Independence for a Contingency Table – Example 18.3
Oij
Support
Against
No Opinion
Total ui
Female 87 32 6
125
.4167
Male 93 70 12
175
.5833
180
102 18
300 j
.60
.34
.06
Eij
Support
Against
No Opinion
Female 75
42.5
7.5
Male
105
59.5
10.5

14. Test of Independence for a Contingency Table – Example 18.4
Null Hypothesis: H0: The two attributes are independent
Alt. Hypothesis: H1: The two attributes are dependent
Test statistic:
Degree of Freedom = (r-1)(c-1) = (2-1)(3-1) = 2
Critical value: .025,2 =
P-value = .0161
Reject H0

15. Test of Independence for a Contingency Table – Example 18.5
Oij
Good
Defective
Total ui
Mac 1
109 11
120 .6
Mac 2 66 14 80 .4
175 25
200 j
.875
.125
Eij
Good
Defective
Mac 1
105 15
Mac 2 70 10

16. Test of Independence for a Contingency Table – Example 18.5
Null Hypothesis: H0: The two attributes are independent
Alt. Hypothesis: H1: The two attributes are dependent
Test statistic:
Degree of Freedom = (r-1)(c-1) = (2-1)(2-1) = 1
Critical value: .01,1 =
P-value = .0809
Fail to reject H0

17. Test of Homogeneity Similar to the test of independence
If row/column totals are fixed, perform a test of homogeneity
Columns 1 2 … c
Rows
O11
O12
O1c u1
O21
O22
O2c u2 r
Or1
Or2
Orc ur 1 2 c

18. Test of Homogeneity
Null Hypothesis: H0: two sets of data are homogeneous
Alt. Hypothesis: H1: sets of data are not homogeneous
Test statistic:
Degree of Freedom = (r-1)(c-1)
Alt. Hypothesis
P-value
Rejection Criterion H1
P(2>02)
02 > 2,df

19. Test of Independence for a Contingency Table – Example 18.6
Oij
Calif. NY
Total ui
Very Satis. 60 75
135
.15
Somewhat Sat.
100
125
225
.25
Somewhat Dissat.
184
140
324
.36
Very Dissatis.
156
216
.24
500
400
900 j
.556
.444
Eij
Calif. NY
Very Satis. 75 60
Somewhat Sat.
125
100
Somewhat Dissat.
180
144
Very Dissatis.
120 96

20. Test of Independence for a Contingency Table – Example 18.6
Null Hypothesis: H0: The two states are homogeneous
Alt. Hypothesis: H1: The two states are not homogeneous
Test statistic:
Degree of Freedom = (r-1)(c-1) = (4-1)(2-1) = 3
Critical value: .025,3 =
P-value = E-9
Reject H0