Lesson 18

This warm-up reminds students of the meanings of mean and MAD by comparing two sets of data with similar but different values and asking whether they will have the same means or MADs or both.

Explain to students that the pairs of data sets are: A and B, X and Y, and P and Q.

For students who have a difficult time starting without calculating, help them to compare the values in the ones place for the first and third pairs of data.

The purpose of the discussion is to bring out methods students used to notice whether the pairs of data sets had the same mean or MAD or both.

Poll the class for each pair of data sets as to whether they had the same mean, MAD, both, or neither.

After students have had a chance to register their vote, ask some students to explain their reasoning for their answer.

In a previous lesson, students compared heights of two teams of people when the entire populations were known. In this activity, students only have access to data from a sample of each population and are asked to determine if the populations are different based on the sample. Students construct informal arguments to explain why the different samples come from populations that are meaningfully different or not (MP3).

Arrange students in groups of 2. Allow students 3–5 minutes quiet work time followed by partner and whole-class discussions.

The purpose of the discussion is for students to think about how comparing groups by using data from samples differs from comparing groups when the population is known.

Ask partners to share their decision about whether the groups had a meaningful difference with the class.

Consider asking these questions for discussion:

• “Compare the information in this activity to the information about team heights given in an earlier lesson.” (In that lesson, we had data from the entire population and here it is only a sample.)
• “Is the overlap of the data more important when you only have a sample or when you have data from the population? Explain your reasoning.” (It is more important when you only have a sample. If there is overlap with only some of the data, it's possible there is more overlap when we include more data from the population.)
• “Is it possible that the data in the two samples were drawn from population data that is identical?” (It is unlikely, but possible.)

The data in the previous activity came from only one sample for each grade. This may not be enough information to make a very good determination about the entire seventh and tenth grade populations. In this activity, students look at different samples from the same population to see that their means are relatively close based on the MADs of the samples. This concept can be reversed to say that if two samples have means that are not very close, then the samples likely came from populations that are quite different. A general rule is given to determine whether two populations are meaningfully different based on the mean and MAD from a sample of each (MP6).

Keep students in groups of 2. Allow students 5 minutes of partner work time, then pause the class to assign samples and explain the general rule.

Ask students to pause after the third question in order to explain the general rule and assign a sample to each group. After all students have paused, assign each group one of the 10 samples to work with for the last 2 questions. Further, explain to students:

• As a general rule, if two populations have the same mean (or median) and similar variability, the sample means (or medians) should be within 2 MADs (or IQRs) of one another.
• If the sample means (or medians) are more than 2 MADs (or IQRs) apart, it is very likely that the population means (or medians) are different. We will say that there is a meaningful difference between the two population means (or medians).
• If the sample means (or medians) are less than or equal to 2 MADs (or IQRs) apart, it is more difficult to say that the two population means (or medians) are very different. In this case we will say that the samples do not provide evidence that the population means (or medians) differ.

Give students 5 more minutes of partner work time followed by a whole-class discussion.

The purpose of the discussion is for students to understand the general rule for determining if two samples suggest a meaningful difference between their populations.

Select at least one group assigned to each of the samples to share their responses to the last 2 questions and record for all to see. Note that all 10 samples from the seventh-grade students have means that are within 2 MADs of one another, but the mean from the tenth-grade student sample is at least 2 MADs away from the mean of each of the seventh grade student samples.

Note that the general rule only has two possible outcomes: “There is a meaningful difference.” or “There is not enough information to say there is a meaningful difference.” If the means are less than 2 MADs apart, the general rule cannot say whether two samples were drawn from populations that contain identical data.

Ask students, “Based only on the dot plots for the 10 samples, would you have guessed that they all might have come from the same population? Explain your reasoning.” (Maybe. There is a lot of overlap among all of the samples.)

In previous lessons, students used sample data to estimate population means and proportions and determined if there is a meaningful difference in population means based on sample means. In this activity, students practice using the general rule developed in the previous activity by estimating the measure of center for a population and comparing populations based on those estimates as well as the associated measure of variability (MP3).

Keep students in groups of 2.

Explain to students that different regions had different raw materials and techniques for constructing metal. One way of testing ancient metal is by looking at the carbon content in the steel. In some cases, this content could determine the region where the metal was made.

Ask students how the general rule from the previous activity might be adapted to use median and interquartile range (IQR) rather than mean and MAD.

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

The purpose of the discussion is for students to understand how to adapt the general rule for determining a meaningful difference between populations to median and IQR.

Consider asking these questions for discussion:

• “Why did this problem use median and IQR instead of mean and MAD?” (Since the distribution for region 1 is not symmetric, it makes more sense to use the median. Also the box plots will show the median and IQR, but there is not a good way to know the mean and MAD.)
• “Is there any overlap in the data from the two regions?” (Yes. The smallest percentage of carbon from the region 1 was well below the median from region 2 while the typical percentage of carbon from region 1 is much greater than from region 2.)
• “On the box plot in the activity, draw a dot two IQRs above the median for region 2. Then draw a star two IQRs below the median for region 1. How do these help you see that there is a meaningful difference in the medians?” (The dot is at 0.53 and the star is at 0.54. Since the median for region 1 is not below the dot nor is the median for region 1 above the star, there must be a meaningful difference.)
• “A piece of steel is found in a place between the two regions sampled. Would testing the percentage of carbon from this metal be useful in determining the region from which it came?” (Yes. Since there is a meaningful difference in the percentage of carbon in the steel from the two regions, it should give a good indication which region created the metal.)