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 explain why some other values might be reasonable or unreasonable based on the sample of data they collected (MP6). They also compare their estimations to the known population proportion and use the class's proportions to gauge the accuracy of their estimate.

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:

• “Why did you select the values that you did for reasonable and unreasonable estimates?” (It is reasonable for estimates to be close to the value that we got. Because we only had 20 values, I think it is reasonable that another estimate would be within about 10 or so of our result. It is unreasonable that the value would be very different from the value we got, so a very large or small number would not be reasonable.)
• “Using the class’s data, how accurate do you think your group's estimate is? Explain your reasoning.” (Answers vary. 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.” (Answers vary. 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. Students must construct an argument for why the character could have the ability to fly using the data to support their argument (MP3).

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “Would you give the ability to fly to any of the new heroes?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback: 

• “ makes sense, but what do you mean when you say . . . ?”
• “What evidence do you have?”
• “How do you know . . . ? What else do you know is true?”

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, ask,

• “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.)
• “Although the proportion of responses from the Mysterious Planets sample who chose flight is only 0.35, it is the most popular choice (freeze, strength, and invisibility split the remaining votes). Does this information help to decide whether to give the new hero the power of flight?” (Yes, because it is the most popular, it might be the best choice. Or it may not be the best choice because more than half of the readers prefer something else. It might help to have another question that directly asks whether they would like the new hero to fly.)
• “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.

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 were 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 and the variability should decrease. In other words, the dots in the dot plot should get closer together towards the center.)