Math 217—Example of Inference in Two-Way Tables

A market research team conducted a survey to investigate the relationship between personality and attitude toward small cars. A sample of 299 adults in a metropolitan area filled out a 16-item self-perception questionnaire, on the basis of which each person was classified into one of three types: cautious conservative, middle-of-the-roader, or confident explorer. The sample of people then gave their overall opinion of small cars: favorable, neutral, or unfavorable. The results of the survey are shown in the two-way table below. Is there a relationship between personality type and attitude toward small cars?

	Personality Type
Attitude	Cautious	Middle-road	Confident
Favorable	79	58	49
Neutral	10	8	9
Unfavorable	10	34	42

The Minitab output (for the chi-square test of independence is shown below):

Chi-Square Test: Attitude Toward Small Cars and Personality Type

Expected counts are printed below observed counts

Chi-Square contributions are printed below expected counts

Cautious Middle-Road Confident Total

Favorable 79 58 49 186

61.59 62.21 62.21

4.924 0.285 2.804

Neutral 10 8 9 27

8.94 9.03 9.03

0.126 0.118 0.000

Unfavorable 10 34 42 86

28.47 28.76 28.76

11.987 0.954 6.092

Total 99 100 100 299

Chi-Sq = 27.289, DF = 4, P-Value = 0.000

Since all expected counts are at least 5 (which is even stronger than our rule-of-thumb for larger than 2x2 tables), we know the chi-squared distribution should be a good approximation for our test statistic (hence, we can trust our p-value). Assuming these two variables are independent, there is essentially no chance (p-value = 0.000) of getting our particular sample data table or a more extreme table. Hence, we have very strong evidence that there is dependence between these variables (that is, there is a significant relationship between these variables).

Now we need to look at appropriate conditional distributions to characterize the nature of the relationship. First look at the Minitab output above. Where are cell counts much larger or smaller than what is expected under independence (see the third row, which lists the chi-square contributions)? In the four outer cells of the table.

Now let’s consider row and columns percentages, respectively:

	Personality Type
Attitude	Cautious	Middle-road	Confident
Favorable	42.5% (r); 79.8% (c)	31.2% (r); 58% (c)	26.3% (r); 49% (c)
Neutral	37.0% (r); 10.1% (c)	29.6% (r); 8% (c)	33.3% (r); 9% (c)
Unfavorable	11.6% (r); 10.1% (c)	39.5% (r); 34% (c)	48.8% (r); 42% (c)

Our significance test showed there is a statistically significant relationship between these two variables. Upon further investigation it looks like cautious people have an overwhelming favorable attitude toward small cars. Whereas, of the personality groups, confident people have the highest unfavorable attitude toward small cars. And, perhaps surprisingly, the highest percentage of confident people actually fall in the favorable category toward small cars. Do you notice other things?