Lesson 19

In this warm-up, students review the useful information that can be gained from different graphical representations of data in preparation for comparing groups based on samples from each.

For classes that may need help remembering the different representations, consider displaying an example of each type of graphical representation mentioned.

Poll the class for their answers to each of the problems. Select at least one student to share their reasoning for each question.

In this info gap activity, students work together to compare two populations from information about samples from each of the populations. Students must pay attention to the information they need in order to solve the problem and the types of question they could ask to get to the answer.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

Tell students they will continue to work with comparing measures of center for populations. Explain the Info Gap structure and consider demonstrating the protocol if students are unfamiliar with it. There are step-by-step instructions in the student task statement.

The purpose of the discussion is to help students understand the types of questions they need to answer in order to compare groups.

Select several groups to share their answers and reasoning for each of the problems. In particular help students understand why the information “The distributions are not symmetric” was important for solving the first problem.

Consider asking these discussion questions:

• “What was the most important question you asked for the first problem? For the second problem?”
• “What are some other ways the information could have been given to solve the problems?” (Instead of the characteristics for the first question, a box plot could have been presented. The second question could have had a dot plot or characteristics like the first problem.)
• “If the distributions for the first problem had been symmetric, would the answer have been the same?” (Yes. The difference in means is 34.25 which is only 1.6 MADs apart, so there would not have been enough information to say that the two population means are meaningfully different.)

In this optional activity, students compare two populations using samples again, but this time only one sample is given. For the other sample, the characteristics have either been computed already or are the focus of the question. This type of analysis is useful when comparing two similar populations as in this activity or when comparing a group against a standard.

Keep students in groups of 2.

Tell students that sometimes it is useful to compare one group to a standard or another group where the important characteristics have already been computed. In these problems, a random sample from one group is given and characteristics of the second group is either given or sought.

Allow students 10 minutes partner work time followed by a whole-class discussion.

The purpose of the discussion is to help students understand how to compare groups when one set of characteristics are known and the other group is represented by sample data.

Select some groups to share their answers and reasoning for the two problems. 

Consider asking these discussion questions:

• “How did you determine what to use as a typical value for the first problem?” (Since there was one value much greater than the others, the distribution would not be symmetric, so median is a more appropriate measure of center.)
• “How did you determine what measure of center to use for the second problem?” (Since the data were all close, either value could be used, but the new equipment reported the “average” or mean, so man should be used for the sample as well.)
• “The manufacturer for the new equipment guarantees 4 flaws or fewer per day with the new equipment. If the new equipment produces only 3 flaws per day does that change the answer for the second problem?” (No. There is an even greater difference between the current and new equipment, so it is even more meaningful.)
• “What other factors would the college graduate want to consider other than the meaningful difference in median salary between the two companies?” (In addition to the other factors for a job such a benefits, relationship with coworkers, type of work being done at each company, etc., the graduate should consider the salary for the type of job he will get at the company. For example, if his degree is in computer science, he may be looking at a job with computers rather than sales or some other department within the company, so he might be able to get a better comparison of salaries that way.)
• “What other factors would the factory manager want to consider other than the meaningful difference in flaws for the equipment?” (The cost of the frequent flaws as well as the cost of the new equipment will probably factor into her decision to buy new equipment. The age of the current equipment and maintenance for older equipment compared to new equipment may also be important.)