CHI-SQUARE TEST OF HOMOGENEITY

Pops P. MacalinoDiscussant

The test for HOMOGENEITY checked if the rows come from the same distribution or appear to come from different distribution

Test of Independence- two categorical variables on a single population

Test of Homogeneity- single categorical variable in two or more

population

Test of Independence Test of Homogeneity

Males and Females

Master’s graduates and Non-MA’s

Tarlaqueῆo, Novo Ecijano, Bulakenyo,

Procter and Gamble, Unilever, Johnson & Johnson

TSU, and CLSU

Republican and Democrat

Example:Suppose that we were to poll registered

voters in reference to charter change. In the plebiscite, 100 voters from rural, 200 from the city were taken by random sampling.

The research question is to determine whether the proportion of voters from each subgroup is the same.

Type of Residence

Opinion on

Favor

Charter Change

OpposeTotal

Rural 80 20 100

City 112 88 200

Total 192 108 300

Ho = There is no difference on the proportion of those who are in

favor of Charter Change in the two groups

Ha = There is a difference on the proportion of those who are in

favor of Charter Change in the two groups

Degrees of Freedom = (c – 1) (r – 1) = (2 – 1) (2 – 1) = 1

Level of Significance (α) = 0.05

Critical Value: (Refer to Appendix E) 3.841 , reject Ho

3.841 , accept Ho

Type of Residence

Opinion on

Favor

Charter Change

OpposeTotal

Rural 80 20 100

City 112 88 200

Total 192 108 300

Contingency TableO E O-E (O-E) ² (O-E)²/E

80.00 64.00 16.00 256.00 4.00

112.00 128.00 -16.00 256.00 2.00

20.00 36.00 -16.00 256.00 7.11

88.00 72.00 16.00 256.00 3.56

∑ = 16.67

Decision:

Since the computed χ² (16.67) is more than the critical value (3.841), the null hypothesis is

rejected.

Conclusion:

The proportion of those who are in favor for charter change is different (not the same) from

the two groups.

Type of Residence

Opinion on

Favor

Charter Change

OpposeTotal

Rural 80 (64) 20 (36) 100

City 112 (128) 88 (72) 200

Total 192 108 300

Sample Problem:Suppose you are interested in knowing whether the

distribution of income classes (low, middle, high) are the same for 200 males and 250 females, at 0.05 level of significance.

LOW INCOME

MIDDLE INCOME

HIGH INCOME

TOTAL

MALE 101 78 21 200

FEMALE 142 73 35 250

TOTAL 243 151 56 450

Ho = There is no difference in the proportion of the distribution of income for males and females.

Ha = There is no difference in the proportion of the distribution of income for males and females.

df : (r-1)(c-1) = (2-1) (3-1) = 2

α = 0.05

cv = 5.991

χ² Hoχ² Ho

Decision: Accept Null Hypothesis since the computed χ² (5.09) is less than the critical value of 5.991

O E O-E (O-E) ² (O-E)²/E

101.00 108.00 -7.00 49.00 0.45

142.00 135.00 7.00 49.00 0.36

78.00 67.11 10.89 118.59 1.7773.00 83.89 -10.89 118.59 1.4121.00 24.89 -3.89 15.13 0.6135.00 31.11 3.89 15.13 0.49

∑ = 5.09

Conclusion: There is no difference on the proportion of the distribution income levels of males and females

VARIABLE - is any characteristics, number, or quantity that can be measured or counted.

Types of Variables

NUMERIC CATEGORICAL

- have values that describe a measurable quantity.

- have values that describe a “quality or characteristic” of a data unit.

Continuous (measurement)

Discrete (countable) ordinal (ranking)

nominal (measures of identity)