Sample Crosstabulation
Problems
Prepared by Dr. H. H. Friedman
A. Please note that the degrees of freedom for the crosstabulation is equal to (#Rows - 1) (#Columns - 1)
B. The expected frequency (theoretical frequency if rows and columns are indeed independent) is equal to the
(row total) x (column total)
n
C. Cramer's coefficient is a measure of association that ranges from 0 to 1. A Cramer's coefficient of 0 indicates that the calculated chi-square is 0, i.e., the observed frequencies are all equal to the expected frequencies. This means that the there is perfect independence between the rows and columns and the column variable provides no information about the row variable. A Cramer's coefficient of 1 indicates that the calculated chi-square is the highest possible chi-square value [n(L-1)]; this indicates a perfect relationship between the rows and columns -- the column variable provides perfect information about the row variable.
(1) Calculate the chi-sqare statistic and the Cramer's coefficient for the following data. Test for significance at the .05 level. The Table value for a Chi-square statistic with 2 degrees of freedom at the .05 level is 5.991.
| Northeast | South | West | |
| Chews Gum | 100 |
70 | 130 |
| Does Not Chew Gum | 300 |
230 | 70 |
[Chi-Square with 2 degrees of freedom = 119 and is significant. Cramer's coefficient is .36]
(2) Calculate the chi-sqare statistic and the Cramer's coefficient for the following data. Test for significance at the .05 level. The Table value for a Chi-square statistic with 2 degrees of freedom at the .05 level is 5.991.
| Region A | Region B | Region C | |
| Prefer Can | 300 |
190 | 60 |
| Prefer Glass Bottle | 200 |
110 | 40 |
[Chi-Square with 2 degrees of freedom = .93 and is not significant. Do not calculate Cramer's coefficient since there is no relationship.]
(3) Calculate the chi-sqare statistic and the Cramer's coefficient for the following data. Test for significance at the .05 level. The Table value for a Chi-square statistic with 2 degrees of freedom at the .05 level is 5.991.
| Protestant | Catholic | Jewish | |
| Been Divorced | 400 |
200 | 100 |
| Never Divorced | 600 |
500 | 200 |
[Chi-Square with 2 degrees of freedom = 24.2 and is significant. Cramer's coefficient is .11.]
(4) Calculate the chi-sqare statistic and the Cramer's coefficient for the following data. Test for significance at the .05 level. The Table value for a Chi-square statistic with 2 degrees of freedom at the .05 level is 5.991.
| Urban | Suburban | Rural | |
| Democrat | 3000 |
1500 | 500 |
| Republican | 2000 |
2500 | 500 |
[Chi-Square with 2 degrees of freedom = 450 and is significant. Cramer's coefficient is .21.]
(5) Calculate the chi-sqare statistic and the Cramer's coefficient for the following data. Test for significance at the .05 level. The Table value for a Chi-square statistic with 2 degrees of freedom at the .05 level is 5.991.
| Latino | African American | Asian American | |
| Prefer Coffee | 150 |
250 | 68 |
| Prefer Tea | 150 |
150 | 132 |
[Chi-Square with 2 degrees of freedom = 44.11 and is significant. Cramer's coefficient is .22.]
(6) Calculate the chi-sqare statistic and the Cramer's coefficient for the following data. Test for significance at the .05 level. The Table value for a Chi-square statistic with 2 degrees of freedom at the .05 level is 5.991.
| Protestant | Catholic | Jewish | |
| Republican | 300 |
110 | 14 |
| Democrat | 200 |
140 | 36 |
[Chi-Square with 2 degrees of freedom = 30.51 and is significant. Cramer's coefficient is .20.]
(7) Calculate the chi-sqare statistic and the Cramer's coefficient for the following data. Test for significance at the .05 level. The Table value for a Chi-square statistic with 3 degrees of freedom at the .05 level is 7.815.
East |
West |
South |
North |
|
| Uses Product | 6 | 29 | 11 | 16 |
| Non-user of Product | 94 | 71 | 89 | 84 |
[Chi-square with 3 degrees of freedom = 22.4 and is significant. Cramer's coefficient is .24]
(8) Calculate the chi-sqare statistic and the Cramer's coefficient for the following data. Test for significance at the .05 level. The Table value for a Chi-square statistic with 2 degrees of freedom at the .05 level is 5.991.
Urban Residents |
Suburban Residents |
Rural Residents |
|
| Democrat | 2000 | 2000 | 700 |
| Republican | 1000 | 4000 | 300 |
[Chi-square with 2 degrees of freedom = 1128 and is significant. Cramer's coefficient is .34]
(9) Calculate the chi-sqare statistic and the Cramer's coefficient for the following data. Test for significance at the .05 level. The Table value for a Chi-square statistic with 2 degrees of freedom at the .05 level is 5.991.
Male |
Female |
|
| Uses Brand A | 32 | 48 |
| Uses Brand B | 42 | 38 |
| Uses Brand C | 26 | 14 |
[Chi-square with 2 degrees of freedom = 7.0 and is significant. Cramer's coefficient is .19]
(10) Calculate the chi-square statistic and the Cramer's coefficient for the following data. The Table value for a Chi-square statistic with 1 degrees of freedom at the .05 level is 3.842. Test at the .05 level to determine whether two appeals used by telemarketers are statistically different in persuading individuals to buy a certain product.
Appeal X -- Telemarketing |
Appeal Y--Telemarketing |
|
| Bought Product | 10 |
40 |
| Did not Buy Product | 90 | 60 |
[Chi-square with 1 degree of freedom = 24 and is significant. Cramer's coefficient is .34]
(11) Calculate the chi-square statistic and the Cramer's coefficient for the following data. Test at the .05 significance level to determine whether a new test to diagnose whether or not an individual has Alzheimer' disease works. The Table value for a Chi-square statistic with 1 degrees of freedom at the .05 level is 3.842.
Tested Positive |
Tested Negative |
|
| Has Alzheimer's disease | 18 |
2 |
| Does not have Alzheimer's disease | 1 | 31 |
[Chi-square with 1 degree of freedom = 40.2 and is significant. Cramer's coefficient is ..87]