Lesson 16

The purpose of this warm-up is for students to compute the fraction of individuals whose responses fall in a specified category. This activity gives students time to think about how to compute these fractions from categorical data.

For the second and third questions, students may debate whether to include the 10 minute times or not. According to the wording of the question asked, it does ask for more than 10 minute times, so maybe exactly 10 minutes should not count (since \(10 \ngtr 10\)). On the other hand, all of the values are listed as whole numbers, so a student who takes 10 minutes and 1 second to get to school may have rounded down to 10, but should have been counted. Noticing the large difference in answers for the third question, it may be worth clarifying the data in this instance, even for an estimate.

Monitor for students who include the 10 minute times for the second and third questions as well as those who do not.

Give students 2 minutes of quiet work time followed by a whole-class discussion.

Select students to share their methods for computing the solutions. Include previously identified students who did or did not include the 10 minute values in their calculations.

If it does not arise during the discussion, explain that answering the third question with the data at hand is only accurate if the sample data is representative of the school. It is possible that the class happens to contain only students who get a ride to school, but much of the school rides the bus.

In previous lessons, students examined the estimation of the mean and median for populations using data from a sample. In this activity, students apply similar reasoning to estimating the proportion of a population that matches certain characteristics. Students collect a sample of 20 reaction times and compute the fraction of responses in their sample that are in a given range. Then, in the discussion, students compare their estimations to the known population proportion and use the class's proportions to gauge the accuracy of their estimate (MP6).

Arrange students in groups of 2. Distribute bags of slips cut from the blackline master.

Tell students that, in statistics, a proportion is a number between 0 and 1 that represents the fraction of the data that fits into the desired category. For example, with the data: {Yes, Yes, Yes, No, Maybe} the proportion of “Yes” answers is \(\frac{3}{5}\) or 0.6.

Introduce the context: All 120 seniors at a high school were asked to click a button as soon as they noticed a box change color and the response time was recorded in seconds. These 120 response times represent the population for this activity. Their responses are written on the slips of paper in the bag.

When selecting a sample of 20, each value does not need to be replaced before taking the next one.

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

The reason a proportion is between 0 and 1 is because we are specifically looking for the proportion of a desired category out of the whole group. This is a “part out of whole” fraction, which will be a number between 0 and 1.

The purpose of the discussion is for students to see how multiple sample proportions can help revise their estimates and give an idea of how accurate the individual estimates from samples might be.

Ask the groups to share the proportion from their sample that had fast reaction times and display the results for all to see.

Some questions for discussion:

• “Using the class’s data, how accurate do you think your group's estimate is? Explain or show your reasoning.” (Students should mention the variability of the proportions from the samples influencing the accuracy of the estimate.)
• “The actual proportion for this population is 0.5. How close was your estimate? Explain why your estimate was not exactly the same.” (Each sample might be slightly different since they do not include all of the values, but they should be close.)
• “If each group had 40 reaction times in their samples instead of 20, do you think the estimate would be more or less accurate?” (The estimate should be more accurate since there is more information available.)

In the previous activity, students collected their own sample and computed an estimate for the population proportion based on the sample. In this activity, students use a different context to practice exploring proportions from samples and their extension to populations. The optional activity following this one continues with this context exploring sampling variability.

Keep students in groups of 2. 

Tell students that three comic books, The Adventures of Super Sam, Beyond Human, and Mysterious Planets, are all planning to add a new superhero to their stories. A survey was sent to dedicated readers of each series to ask what type of ability the new hero should have: fly, freeze, or another power.

Display the tables from the Task Statement for all to see. Ask students, “What do you notice? What do you wonder?”

Students may notice:

• There are 4 different responses: fly, freeze, super strength, and invisibility.
• The number of responses for each of the 4 different responses.
• There are 20 responses.

Students may wonder:

• Will the decision for the new hero's power be based only on this survey?
• What proportion chose each of the different powers?
• Is this sample of 20 responses representative of the population?

Give students 5–7 minutes of partner work time followed by a whole-class discussion.

The purpose of the discussion is for students to recognize the value of a proportion and why a sample may be necessary to estimate the proportion for the population.

Some questions for discussion:

• “Explain why you think a sample was used instead of the population for this situation.” (The population is too large to ask all of the readers about their preference. Also, the authors may want the power of the new hero to be a surprise to some readers, so they want to get some information without telling everyone about what is coming.)
• “Based on the proportions computed in each of these samples, would you suggest to the authors of any of the comic books to give their new hero the ability to fly?”
• “Although the proportion of responses from the Mysterious Planets sample who chose flight was only 0.35, it was the most popular choice (freeze, super strength, invisibility, and other powers split the remaining votes). Does this information change your answer to the previous question?”
• “Of the three estimates, which do you think is most accurate? Explain your reasoning.” (Beyond Human has the largest sample, so it may be the most accurate.)

This optional activity goes beyond grade-level expectations to deepen students’ understanding of sampling variability.

This activity continues the comic book context introduced in the previous activity. There is not a measure of variability such as MAD or IQR for proportions since the data are categorical rather than quantitative, so other methods must be employed to determine the accuracy of an estimate. Students look at dot plots showing the results from multiple samples to gauge the accuracy of the estimates for population proportions (MP7). This activity will provide a foundation for work in later grades.

Arrange students in groups of 2.

Help students make sense of the dot plot. Each dot in the dot plot represents the proportion from a random sample of 20 readers. 50 random samples were taken and the 50 proportions are plotted on the dot plot. For example, the 0.4 proportion from the previous activity would be represented by one of the 4 dots at 0.4 on the first dot plot.

Ask, “Are any of the sample proportions greater than or equal to 0.5? What does this mean?” (Yes, 2 dots are greater than or equal to 0.5. This means that, in those samples, at least half of the people prefer the new hero to have the ability to fly.)

Give students 5–7 minutes of partner work time followed by a whole-class discussion.

The purpose of the discussion is for students to talk about how the variability in sample proportions affects their trust in the estimates of the population proportion.

Consider these questions for discussion:

• “When estimating a population mean or median from a random sample, measures of variability from a sample can be used to help gauge the accuracy of the estimate. With proportions there is not a measure of variability in the same way. How did the information in this activity guide your thoughts about the accuracy of the population estimate?” (Many samples were taken and their proportions computed. The variability of these sample proportions showed how much to trust the population estimate.)
• “How does the distribution of values in the dot plots of sample proportions affect your trust in an estimate of population proportion?” (The more variability, the less certainty in the estimate.)
• “How would the distributions change if the number of responses in each sample were increased?” (The center should remain about the same, the variability should decrease or, in other words, the dots in the dot plot should get closer together towards the center.)