Chapter 14 – 1

Chi-Square• Chi-Square as a Statistical Test• Statistical Independence• Hypothesis Testing with Chi-Square

• The Assumptions• Stating the Research and Null Hypothesis• Expected Frequencies• Calculating Obtained Chi-Square• Sampling Distribution of Chi-Square• Determining the Degrees of Freedom

• Limitations of Chi-Square Test

Chapter 14 – 2

Chi-Square as a Statistical Test• Chi-square test: an inferential statistics technique

designed to test for significant relationships between two variables organized in a bivariate table.

• Chi-square requires no assumptions about the shape of the population distribution from which a sample is drawn.

• It can be applied to nominally or ordinally measured variables.

Chapter 14 – 3

Gender and Fear of Walking Alone at Night

SEXMale Female Total

No 21(87%) 14(40%) 35(59%)

AreYouAfraidToWalkAlone

Yes 3(13%) 21(60%) 24(41%)

24(100%) 35(100%) 59(100%)Gender and Fear are associated but can this relationship be generalized to the population?

Chapter 14 – 4

Statistical Independence

• Independence (statistical): the absence of association between two cross-tabulated variables. The percentage distributions of the dependent variable within each category of the independent variable are identical.

Chapter 14 – 5

Gender and Fear of Walking Alone at Night

SEXMale Female Total

No 14(59%) 21(59%) 35(59%)

AreYouAfraidToWalkAlone

Yes 10(41%) 14(41%) 24(41%)

24(100%) 35(100%) 59(100%)Gender and Fear are not related

Chapter 14 – 6

Hypothesis Testing with Chi-SquareChi-square follows five steps:

1. Making assumptions

2. Stating the research and null hypotheses and selecting alpha

3. Selecting the sampling distribution and specifying the test statistic

4. Computing the test statistic

5. Making a decision and interpreting the results

Chapter 14 – 7

The Assumptions• The chi-square test requires no assumptions

about the shape of the population distribution from which the sample was drawn.

• However, like all inferential techniques it assumes random sampling.

• It can be applied to variables measured at a nominal and/or an ordinal level of measurement.

Chapter 14 – 8

Stating Research and Null Hypotheses

• The research hypothesis (H1) proposes that the two variables are related in the population.

• The null hypothesis (H0) states that no association exists between the two cross-tabulated variables in the population, and therefore the variables are statistically independent.

Chapter 14 – 9

H1: The two variables are related in the population.

Gender and fear of walking alone at night are statistically dependent.

H0: There is no association between the two variables.

Gender and fear of walking alone at night are statistically independent.

Chapter 14 – 10

The Concept of Expected FrequenciesExpected frequencies fe : the cell frequencies that would be expected in a bivariate table if the two tables were statistically independent.

Observed frequencies fo: the cell frequencies actually observed in a bivariate table.

Chapter 14 – 11

Calculating Expected Frequencies

fe = (column marginal)(row marginal)N

To obtain the expected frequencies for any cell in any cross-tabulation in which the two variables are assumed independent, multiply the row and column totals for that cell and divide the product by the total number of cases in the table.

Chapter 14 – 12

Gender and Fear -Observed and Expected Frequencies

SEXMale Female Total

No 21(1144) 14(2211) 35

AreYouAfraidToWalkAlone

Yes 3 (1100) 21(1144) 24

Total 24(100%) 35(100%) 59(100%)((24)*(35))/59=14

Chapter 14 – 13

Calculating the Obtained Chi-Square

e

oe

fff 2

2 )(

fe = expected frequenciesfo = observed frequencies

Chapter 14 – 14

Calculating the Obtained Chi-Square

Fo Fe Fo-fe (Fo-fe)2 (fo-fe)2/feMaleNo

21 14 7 49 3.5MaleYes

3 10 -7 49 4.9FemaleNo

14 21 -7 49 2.3FemaleYes

21 14 7 49 3.5 14.2

2.14)( 22

fe

fefo

Chapter 14 – 15

The Sampling Distribution of Chi-Square

• The sampling distribution of chi-square tells the probability of getting values of chi-square, assuming no relationship exists in the population.

• The chi-square sampling distributions depend on the degrees of freedom.

• The sampling distribution is not one distribution, but is a family of distributions.

Chapter 14 – 16

The Sampling Distribution of Chi-Square

• The distributions are positively skewed. The research hypothesis for the chi-square is always a one-tailed test.

• Chi-square values are always positive. The minimum possible value is zero, with no upper limit to its maximum value.

• As the number of degrees of freedom increases, the distribution becomes more symmetrical.

Chapter 14 – 17

Chapter 14 – 18

Determining the Degrees of Freedom

df = (r – 1)(c – 1)

wherer = the number of rowsc = the number of columns

Chapter 14 – 19

Calculating Degrees of FreedomHow many degrees of freedom would a table with 3 rows and 2 columns have?

(3 – 1)(2 – 1) = 2 2 degrees of freedom

Chapter 14 – 20

The Sampling Distribution of Chi-square-Appendix D

df .05 .01 .001

1 3.841 6.635 10.827

2 5.991 9.210 13.815

3 7.815 11.341 16.268Our obtained Chi Square > than 10.827 with p =.001. Thus our Chi-Square would occur even less than that if H0 was true. We reject H0 at the .001 level.

Chapter 14 – 21

Limitations of the Chi-Square Test• The chi-square test does not give us much

information about the strength of the relationship or its substantive significance in the population.

• The chi-square test is sensitive to sample size. The size of the calculated chi-square is directly proportional to the size of the sample, independent of the strength of the relationship between the variables.

• The chi-square test is also sensitive to small expected frequencies in one or more of the cells in the table.