Musical Preferences

Task

The 54 students in one of several middle school classrooms were asked two questions about musical preferences: “Do you like rock?” “Do you like rap?” The responses are summarized in the table below.

1. Is this a random sample, one that fairly represents the opinions of all students in the middle school?
2. What percentage of the students in the classroom like rock?
3. Is there evidence in this sample of a positive association in this class between liking rock and liking rap? Justify your answer by pointing out a feature of the table that supports it.
4. Explain why the results for this classroom might not generalize to the entire middle school.

IM Commentary

There is a variety of approaches to (c) and answers may vary considerably. The basic idea is for students to demonstrate that they know what it means for two variables to be associated: that if we knew someone were in one group (for example, they like rap), we now know more about their preferences for rock than if we knew nothing at all.

A productive follow-up discussion is to ask students what sort of numbers they would see in the table if there were no association. If there were no association, we'd see that about 60% of the students like rap in both the "like rock" and "do not like rock" groups. A common mistake is for students to think that if there is no association, the overall percentages of those who like rock must be the same as the overall percentage of those who like rap. Another common mistake is for students to think that a lack of association means that all percentages must be 50%.

Students will wonder how close the percentages must be to conclude that there is no association. For example, suppose 61% of those who like rap like rock, and 60% of those who do not like rap like rock. Is this "close enough" to conclude there is no association? This is a question that is answered when students learn about inference and learn to compare two proportions. Because there is no hard-and-fast rule, the question is phrased to ask whether there is evidence of an association, and does not ask whether there is an association.

Note that if there were more than three categories for responses in either of the two variables, (for instance, "likes rap", "does not like rap", "no preference"), the question is more complicated because more categories must be considered.

Solution

1. This is not a randomly selected sample that fairly represents the students in the school. See part (d) for more details.
2. 33/54 = 61.1%
3. Yes, there is evidence of a positive association. Of those who like Rap, 27/31 = 87.1% like Rock, too. This means that the percentage of those who like Rock is higher among those who like Rap than among the entire sample.
4. The sample is not necessarily a random sample. While it might be true that the association holds in other classes, we have no evidence of this. It is possible, for instance, that this was an unusual class at this school; maybe this class consisted entirely of music students, and their preferences would be different than in other classes or than in the entire school.