{"id":577,"date":"2024-10-18T02:51:29","date_gmt":"2024-10-18T02:51:29","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/mat1260\/?post_type=chapter&p=577"},"modified":"2025-01-28T23:33:16","modified_gmt":"2025-01-28T23:33:16","slug":"11-2-chi-squared-test-for-independence","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/mat1260\/chapter\/11-2-chi-squared-test-for-independence\/","title":{"raw":"11.2: Chi-Squared Test for Independence","rendered":"11.2: Chi-Squared Test for Independence"},"content":{"raw":"
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Overview<\/h2>\r\n

The last three procedures that we studied (two-sample t, paired t, and ANOVA) all involve the relationship between a categorical explanatory variable and a quantitative response variable, corresponding to Case C\u2192Q in the role\/type classification table below. Next, we will consider inferences about the relationships between two categorical variables, corresponding to case C\u2192C.<\/p>\r\n\"It<\/span><\/span>\r\n

In the Exploratory Data Analysis unit of the course, we summarized the relationship between two categorical variables for a given data set (using a two-way table and conditional percents), without trying to generalize beyond the sample data.<\/p>\r\n

Now we perform statistical inference for two categorical variables, using the sample data to draw conclusions about whether or not we have evidence that the variables are related in the larger population from which the sample was drawn. In other words, we would like to assess whether the relationship between X and Y that we observed in the data is due to a real relationship between X and Y in the population or if it is something that could have happened just by chance due to sampling variability.<\/p>\r\n\"We<\/span><\/span>\r\n

The statistical test that will answer this question is called the\u00a0chi-square test for independence<\/em>. Chi is a Greek letter that looks like this: [latex]\\chi[\/latex], so the test is sometimes referred to as: The [latex]\\chi ^{2}[\/latex] test for independence.<\/p>\r\n

The structure of this section will be very similar to that of the previous ones in this module. We will first present our leading example, and then introduce the chi-square test by going through its 4 steps, illustrating each one using the example. We will conclude by presenting another complete example. As usual, you\u2019ll have activities along the way to check your understanding, and to learn how to use software to carry out the test.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Let\u2019s start with our leading example.<\/p>\r\n\r\n

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Example<\/h3>\r\n<\/header>\r\n
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In the early 1970s, a young man challenged an Oklahoma state law that prohibited the sale of 3.2% beer to males under age 21 but allowed its sale to females in the same age group. The case (Craig v. Boren, 429 U.S. 190, 1976) was ultimately heard by the U.S. Supreme Court.<\/p>\r\n

The main justification provided by Oklahoma for the law was traffic safety. One of the 3 main pieces of data presented to the court was the result of a \u201crandom roadside survey\u201d that recorded information on gender, and whether or not the driver had been drinking alcohol in the previous two hours. There were a total of 619 drivers under 20 years of age included in the survey.<\/p>\r\n

Here is what the collected data looked like:<\/p>\r\n\"A<\/span><\/span>\r\n

The following two-way table summarizes the observed counts in the roadside survey:<\/p>\r\n\"A<\/span><\/span>\r\n

Our task is to assess whether these results provide evidence of a significant (\u201creal\u201d) relationship between gender and drunk driving.<\/p>\r\n

The following figure summarizes this example:<\/p>\r\n\"The<\/span><\/span>\r\n

Note that as the figure stresses, since we are looking to see whether drunk driving is related to gender, our explanatory variable (X) is gender, and the response variable (Y) is drunk driving. Both variables are two-valued categorical variables, and therefore our two-way table of observed counts is 2-by-2. It should be mentioned that the chi-square procedure that we are going to introduce here is not limited to 2-by-2 situations, but can be applied to any r-by-c situation where r is the number of rows (corresponding to the number of values of one of the variables) and c is the number of columns (corresponding to the number of values of the other variable).<\/p>\r\n

Before we introduce the chi-square test, let\u2019s conduct an exploratory data analysis (that is, look at the data to get an initial feel for it). By doing that, we will also get a better conceptual understanding of the role of the test.<\/p>\r\n

Exploratory Analysis<\/em><\/p>\r\n

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 \u2014 the percentages of the response variable for each value of the explanatory variable \u2014 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.<\/p>\r\n

Here is the table of conditional percentages:<\/p>\r\n\"A<\/span><\/span>\r\n

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:<\/p>\r\n\r\n