12.1 Exploratory Analysis

Recall that the key to reporting appropriate summaries for a two-way table is deciding which of the two categorical variables plays the role of explanatory variable, and then calculating the conditional percentages — the percentages of the response variable for each value of the explanatory variable — separately. In this case, since the explanatory variable is gender, we would calculate the percentages of drivers who did (and did not) drink alcohol for males and females separately.

Here is the table of conditional percentages:

TA <- xtabs(~ Gender + DroveDrunk, data = DF)
prop.table(TA, 1)

        DroveDrunk
Gender          No       Yes
  Female 0.8840580 0.1159420
  Male   0.8399168 0.1600832

For the 619 sampled drivers, a larger percentage of males were found to be drunk than females (16.0% vs. 11.6%). Our data, in other words, provide some evidence that drunk driving is related to gender; however, this in itself is not enough to conclude that such a relationship exists in the larger population of drivers under 20. We need to further investigate the data and decide between the following two points of view:

• The evidence provided by the roadside survey (16% vs 11.6%) is strong enough to conclude (beyond a reasonable doubt) that it must be due to a relationship between drunk driving and gender in the population of drivers under 20.

• The evidence provided by the roadside survey (16% vs. 11.6%) is not strong enough to make that conclusion, and could have happened just by chance, due to sampling variability, and not necessarily because a relationship exists in the population.

Actually, these two opposing points of view constitute the null and alternative hypotheses of the chi-square test for independence, so now that we understand our example and what we still need to find out, let’s introduce the four-step process of this test.