A marketing manager wants to find out whether owning a car is associated with family size for a particular state. He surveys 100 families in the said state.
The dependent variable is family size, the family size would be identified as small (2 or 3 members only), average (4-5 members only) and large (more than 5 members). Family size is an ordinal data since the values are arranged from small, average and large. Family size can be measured by sending out a product questionnaire to the participants through mail or at local grocery stores.
The independent variable is having a car or not, the responses will only be limited to yes (own a car) and no (don’t own a car). Which is a nominal data with assigned values as 0 = no car and 1 = yes (have a car). This can be surveyed together with the product questionnaire.
Is car ownership associated with family size?
Null hypothesis: Car ownership and family size is independent of each other.
Alternative hypothesis: Car ownership and family size is not independent (they are related to each other).
The chi-square statistics is a nonparametric test because it has no assumptions, it does not test whether one variable is greater than the other or that the independent variable causes the change in the dependent variable. The chi-square test is appropriate for this problem because the measured variables meet the requirements of the chi-square, an ordinal dependent variable (family size) and a nominal independent variable (car ownership). Likewise, the samples need not be normally distributed. Statistical tests like the chi-square can be used in very large samples of respondents; wherein the data is reported in numerical values which should not be less than 5 and uses a contingency table in determining the expected and actual values of the gathered data (Sirkin,1995).
Tests of associations cannot be measured by ANOVA or correlation because the variables do not meet the requirements of the said statistical tools. ANOVA can only be used if there are two or more categories for each variable which should be all ordinal or all nominal. Correlation although can establish relationships can only be used when there is a linear relationship between the variables, and in this problem there is no way of telling that. Lastly, the problem only tests for association not effect or the direction of the association or relationship.
Since the chi-square is limited into testing whether two variables are independent of each other or whether they are associated and does not test for strength of the association, hence effect size is not possible in this test.
Sirkin, R. M. (1995). Statistics for the social sciences. California: Sage Publications.