# Lesson 11

Reducing Margin of Error

## 11.1: Notice and Wonder: Female Leads (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit the idea that an increase in sample size is associated with a decrease in margin of error, which will be useful when students investigate the relationship between margin of error and sample size in a later activity. While students may notice and wonder many things about these images, the relationship between sample size and margin of error is the important discussion point. This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that the margin of error decreases as the sample size increases.

### Launch

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the table for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

### Student Facing

Five students wanted to see how many children’s movies have female lead characters. They each took a sample of children’s movies, found the proportion of movies that had female lead characters, then used their results to simulate 100 additional samples. The table shows some of the findings based on the original sample and the simulations.

What do you notice? What do you wonder?

student number of movies used in the original random sample estimated proportion margin of error
A 20 0.273 0.204
B 20 0.205 0.189
C 30 0.280 0.180
D 30 0.232 0.157
E 50 0.205 0.115

### Activity Synthesis

The purpose of the discussion is to notice that the margin of error tends to decrease when a greater number of things are used in the sample. Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the table. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification or point out contradicting information. If the relationship between the margin of error and sample size does not come up during the conversation, ask students to discuss this idea.

## 11.2: Finding a Job (10 minutes)

### Activity

The mathematical purpose of this activity is for students to begin to understand the relationship between sample size and margin of error. Two characters collect samples and notice that their simulations result in very different margins of error. Students should begin to notice that a larger sample size should result in an estimate with a lesser margin of error.

### Launch

Reading: MLR6 Three Reads. Use this routine to support reading comprehension of this problem. Ask students to keep their books or devices closed and display only the task statement without revealing the questions that follow. Use the first read to orient students to the situation. After a shared reading, ask, “What is this situation about?” (two students found a proportion from a random sample). After the second read, students list any quantities that can be counted or measured, without focusing on specific values (proportions, margins of error, and number of simulations). During the third read, the question or prompt is revealed. Invite students to discuss possible strategies to answer the questions, referencing the relevant quantities named after the second read. This helps students connect the language and reasoning needed to solve a problem while maintaining the cognitive demand of the task.
Design Principle: Support sense-making
Representation: Internalize Comprehension. Begin by providing students with questions to help them form generalizations about mean, sample size, standard deviation, and margin of error. For example, “When the mean changes, does the margin of error change? When the standard deviation changes, does the margin of error change?”
Supports accessibility for: Conceptual processing

### Student Facing

Elena and Clare are each working on a project about how high school students are having trouble finding jobs. They each find the proportion of students without jobs from a random sample, then use a computer to do 1,000 simulations using the proportion they found and report the results.

Elena says, “The proportion of high school students without jobs is about 0.70 with a margin of error of 0.280.”

Clare says, “The proportion of high school students without jobs is about 0.74 with a margin of error of 0.138.”

1. Both students reported the margin of error based on 2 standard deviations from their simulations. What are the mean and standard deviation each student found? For at least one student, show your reasoning.
2. Clare and Elena try to figure out why Clare had such a smaller range of values in her report.
1. First they consider the proportion they used in the simulations. Elena says, “My simulation used 0.7 as the proportion since I found that proportion in my original sample.” Clare says, “My simulation used 0.75 as the proportion since I found that proportion in my original sample.” The students used different proportions in their simulations. Do you think this is why Clare has a smaller margin of error? Explain your reasoning.
2. They look for more differences in their initial sample and discover than Elena surveyed 10 people in her initial sample and Clare surveyed 40 people. Do you think this is why Clare has a smaller margin of error? Explain your reasoning.

### Activity Synthesis

The purpose of this discussion is for students to think about the relationship between sample size and margin of error. Ask students,

• “Why do you think Clare continues to get a smaller margin of error than Elena?” (She took a bigger sample so the samples are more likely to be representative of the population.)
• “Elena had a sample size of 10 and Clare had a sample size of 40. Why do you think the sample size could have caused Clare’s margin of error to be less than Elena’s margin of error?” (Since there are more people included in each sample, the samples themselves capture more of the population variability, so each sample estimate is more likely to accurately reflect the population characteristic.)

## 11.3: Exercised (20 minutes)

### Activity

The mathematical purpose of this activity is for students to collect and analyze data to further their conceptual understanding of the relationship between sample size and margin of error. Students repeatedly collect samples of various sizes to estimate a proportion and margin of error for data about the class. Students should notice that a greater sample size results in a lesser margin of error.

### Launch

Arrange students in groups of 5. Give each student a bag. Assign each group a number of items to include in their sample: 5, 10, or 15. Provide access to devices that can run GeoGebra or other statistical technology.

After students have completed the first question, select one student from each group to report the number of slips in each sample, the mean of the 50 proportions, the standard deviation of the 50 proportions, as well as the margin of error. Display the results for all to see.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Students can work with a partner to complete the trials more efficiently (one person drawing paper from bag, another writing the results).
Supports accessibility for: Language; Social-emotional skills

### Student Facing

Cut a sheet of paper into enough slips for each student in the class to get one. On each of the slips you cut, write “Yes” if you spend at least 5 hours intentionally exercising each week, otherwise write “No” on each of the slips. Put one of your paper slips in each student’s bag, including your own.

After all the slips are distributed for all the students, return to your bag. Your teacher gave your group a number of slips to draw for each sample. Draw a sample and record the proportion of the slips that say “Yes.” Return the slips to the bag and repeat the process until you have 10 sample proportions from 10 samples. Share your results with the group so that each person has 50 sample proportions to work with.

1. Use your 50 sample proportions to report an estimate and associated margin of error for the class. Explain or show your reasoning.
2. Compare the standard deviations of the 50 sample proportions for each of the different groups. Is there a connection to the number of slips chosen in each sample?

### Student Facing

#### Are you ready for more?

Here is a table of standard deviations that were obtained when repeatedly sampling from a population where the proportion of “Yes” slips was 0.4.

sample size standard deviation
5 .219
20 .110
45 .073
80 .055
125 .044
500

Estimate the standard deviation for a sample size of 500.

### Anticipated Misconceptions

Some students may wish to select multiple samples at the same time by selecting extra pieces of paper at once rather than replacing the papers after each sample. Ask students to reflect on why that process does not produce representative samples. In particular, remind students that randomness is essential for selecting unbiased samples and selecting multiple samples at once means the samples are biased (because a paper can not be included in both samples using this method).

### Activity Synthesis

The goal of this discussion is for students to understand that as the sample size increases the margin of error decreases.

Here are some questions for discussion:

• “What was the sample size for the 50 sample proportions that had the lowest standard deviation?” (The groups that did 15 samples had the lowest standard deviation.)
• “What was the sample size for the 50 sample proportions that had the highest standard deviation?” (The groups that did 5 samples had the highest standard deviation.)
• “What conjectures do you have about the sample size and the standard deviation?” (I think that a larger sample size decreases the standard deviation of the sample proportions. That also means that as the sample size increases, the margin of error decreases.)
• “Why do you think an increase in sample size tends to result in a decrease in variability? Explain your thinking.” (The bigger sample sizes tend to be less variable because the larger sample sizes more accurately represent the population, so each sample proportion is likely to be closer to the population proportion than the sample proportions from smaller samples.)
• “What is the relationship between sample size and margin of error? Explain your reasoning.” (As the sample size increases the margin of error decreases. I know that a large sample will likely result in a lower standard deviation than a smaller sample. Since the margin of error is directly related to the standard deviation, I know that the margin of error will likely decrease.)
Speaking: MLR8 Discussion Supports. Use this routine to prepare students for the whole-group discussion. At the appropriate time, give groups 2–3 minutes to plan what they will say when they present their responses to the last question. Encourage students to consider what details are important to share and to think about how they will explain their reasoning using mathematical language.  Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation

## Lesson Synthesis

### Lesson Synthesis

Here are some questions for discussion:

• “Han runs a simulation 500 times using a sample size of 20. Mai runs a similar simulation 500 times using a sample size of 50. When Han and Mai use their data to estimate a population proportion, who do you think will report a higher margin of error? Explain your reasoning.” (Han will report a larger margin of error because Han’s sample size was smaller, so the standard deviation is likely to be greater than Mai’s.)
• “Han calculates the mean, 0.51, and standard deviation, 0.14, of the sample proportions. If he uses this data to estimate the population proportion, what is a reasonable estimate for the margin of error?” (0.28)
• “Explain what is means to estimate a population proportion with a mean of 0.51 and a margin of error of 0.48.” (It means that there is a 95% chance the actual population proportion is between 0.23 and 0.79 because that interval is the mean plus or minus the margin of error.)

## Student Lesson Summary

### Student Facing

Estimating a population characteristic from a random sample will always have some room for error since the sample is only a subset of the population, so it does not provide complete information about the population. One way to reduce the error is to take larger random samples. Not only will this include more of the population in the sample, so a greater percentage of the total information is being recorded, but the standard deviation of a sample statistic from simulations using larger samples also tends to be smaller. With a smaller standard deviation, the difference between the sample estimate and the actual value of the population characteristic being estimated tends to be smaller.

For example, a group goes to an island and collects a random sample of 10 lizards, finding that 5 of them are male. This random sample has a proportion of 0.5 males in the group. How close is this likely to be to the actual proportion of lizards that are male on the island? To investigate, we can simulate additional samples from a population in which the proportion is 0.5 and see how far away from 0.5 the sample proportions tend to be. The distribution of simulated samples will give us an idea of how far off our sample estimate of 0.5 might be from the actual population value.

Suppose we simulate taking 30 random samples of 10 lizards from a population with a 0.5 probability of each one being male, and this results in sample proportions that have a mean of 0.503 and a standard deviation of 0.145. The dot plot of the sample proportions is approximately normal in shape, so it is reasonable to think that simulated proportions from the population should be within about 2 standard deviations, or 0.290 ($$2 \boldcdot 0.145 = 0.290$$) of the actual population mean.

Based on the simulations and analysis, we expect that the original estimate of 0.5 for the proportion of the population that is male is likely to be within 0.290 of the actual value of the population proportion of lizards that are male. The researchers should report an estimate of 0.5 for the population proportion with an associated margin of error of 0.290.

Later, another group goes to the island and collects a sample of 40 lizards, finding that 20 of them are male. After simulating 30 samples of 40 lizards with a 0.5 probability of each one being male, the mean proportion that is male is found to be 0.503 again, but the standard deviation is 0.062. This group should report an estimated proportion of lizards on the island that are male of 0.503 with a margin of error of 0.124 ($$2 \boldcdot 0.062 = 0.124$$). This means that they believe their estimate of 0.503 is within 0.124 of the actual population proportion.

Although the means are the same in this case, the standard deviation is much less with the larger samples, so the margin of error reported was smaller.