Lesson 19

Comparing Populations With Friends

19.1: Features of Graphic Representations (5 minutes)

Warm-up

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.

Launch

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

Student Facing

Dot plots, histograms, and box plots are different ways to represent a data set graphically.

Which of those displays would be the easiest to use to find each feature of the data?

1. the mean
2. the median
3. the mean absolute deviation
4. the interquartile range
5. the symmetry

Activity Synthesis

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

19.2: Info Gap: Comparing Populations (30 minutes)

Activity

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:

Launch

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.

Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to use data from samples to compare two populations. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

3. Explain how you are using the information to solve the problem.

Continue to ask questions until you have enough information to solve the problem.

4. Share the problem card and solve the problem independently.

If your teacher gives you the data card:

2. Ask your partner “What specific information do you need?” and wait for them to ask for information.

If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

4. Read the problem card and solve the problem independently.

5. Share the data card and discuss your reasoning.

Student Facing

Is there a meaningful difference between top sports performance in two different decades? Choose a variable from your favorite sport (for example, home runs in baseball, kills in volleyball, aces in tennis, saves in soccer, etc.) and compare the leaders for each year of two different decades. Is the performance in one decade meaningfully different from the other?

Activity Synthesis

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.

• “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.)

19.3: Comparing to Known Characteristics (15 minutes)

Optional activity

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.

Launch

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.

Student Facing

1. A college graduate is considering two different companies to apply to for a job. Acme Corp lists this sample of salaries on their website:

$45,000$55,000

$140,000$70,000

$60,000$50,000

What typical salary would Summit Systems need to have to be meaningfully different from Acme Corp? Explain your reasoning.

2. A factory manager is wondering whether they should upgrade their equipment. The manager keeps track of how many faulty products are created each day for a week.
• 6
• 7
• 8
• 6
• 7
• 5
• 7
The new equipment guarantees an average of 4 or fewer faulty products per day. Is there a meaningful difference between the new and old equipment? Explain your reasoning.

Activity Synthesis

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.

• “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.)
Representation: Internalize Comprehension. Use color and annotations to illustrate connections between representations. As students share their diagrams and reasoning, use color and annotations to scribe their thinking on a display of each problem so that it is visible for all students.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each response or observation that is shared, ask students to restate and/or revoice what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker whether their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped clarify the original statement. This will provide more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

Ask students what information is important to collect when attempting to compare large groups and why each of these pieces of information is useful. Ask students if they can think of other situations in which it might be helpful to compare two large groups by generating a sample and collecting information.

Student Lesson Summary

Student Facing

When using samples to comparing two populations, there are a lot of factors to consider.

• Are the samples representative of their populations? If the sample is biased, then it may not have the same center and variability as the population.
• Which characteristic of the populations makes sense to compare—the mean, the median, or a proportion?
• How variable is the data? If the data is very spread out, it can be more difficult to make conclusions with certainty.

Knowing the correct questions to ask when trying to compare groups is important to correctly interpret the results.