Example. Men, Women, and the Relative Importance of Personality versus Looks.
In our January 14 class survey, a questions was
In deciding whether to date someone, rate the relative importance of personality
versus looks on a scale of 0 (personality most important) to 9 (looks most
important).
I created three categories for the “relative importance” responses: 0 to 3, 4 to 5, and 6 to
9. Table 14.1 compares the 68 men and 136 women in the survey with regard to these
categories. Within a cell, the top number is a count while the number below the count is
the row percent for that category.
Table 14.1 Gender and Relative Importance of Personality versus Looks
female
male
0-3
31
22.79%
7
10.29%
Relative Importance
4-5
6-9
85
20
62.50%
14.71%
42
61.76%
19
27.94%
All
136
68
Figure 14.1 is a bar chart of the row percentages. This graph clearly shows a different
pattern of responses for mean and women.
Figure 14.1
A Chi-square procedure can be used to analyze the statistical significance of the
relationship between two categorical variables. Generally, the null hypothesis is that
there is not a relationship. Within a specific problem, we may write a more elaborate
version of this null hypothesis. For instance, in our example the null and alternative
hypotheses could be expressed as:
H0 : No difference between men and women with regard to the distribution of
responses in the “Importance Categories” (in other words, relative importance of
looks and sex are not related)
Ha : There is a difference between men and women
Table 14.2 shows the Minitab output for the Chi-square test of these hypotheses. We see
that the p-value of the Chi-square test is 0.019. This is a “statistically significant” result
so we can declare a difference between men and women. This is a conclusion that applies
to the populations of men and women represented by the samples from our class.
Table 14.2 Chi-square Test for Relationship Between Gender and Relative
Importance of Personality versus Looks.
Relative Importance
0-3
4-5
6-9
31
85
20
25.33
84.67
26.00
All
136
136.00
male
7
12.67
42
42.33
68
68.00
All
38
38.00
127
127.00
female
19
13.00
39
39.00
204
204.00
Chi-Square = 7.960, DF = 2, P-Value = 0.019
Some Technical Details of the Chi-Square Procedure
Ø The null hypothesis is that there is no relationship between two categorical variables.
Ø The Chi-square statistic measures the difference between the observed counts in the
cells of the two-way table and a set of “expected” counts.
Ø The expected counts are hypothetical counts that would occur if the null hypothesis
were true.
Calculation of Expected Counts
For each cell in a two-way table,
Expected Count =
Row Total × Column Total
Total n
Example: For the “female, 0-3” cell in our example
Expected Count =
136 × 38
= 25.33
204
Ø The Chi-square statistic is
Χ2 =
∑
( Observed - Expected) 2
∑
Expected
all cells
stands for “sum” and the sum is over all cells of the table.
Ø If the null hypothesis is true, the p-value is approximately the area to the right of the
calculated Chi-square value in a “Chi-Square probability distribution” that has
degrees of freedom = (rows – 1)(columns – 1). This is the probability that the Chisquare would be as large as it is if the null hypothesis were true.
In our example, there are two rows and three columns so the df= (2-1)(3-1)=2. The pvalue is the area to the right of the calculated Chi-square of 7.96 (see Table 14.2). This
area is the probability that the Chi-square would be 9.96 or large if the null hypothesis is
true. The answer, p-value-0.019, tells us that we can reject the null hypothesis.
Figure 14.2 illustrates the p- value for our example.