# Lesson 10

Estimating Proportions from Samples

## 10.1: Math Talk: Proportions (5 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings that students have for working with proportions. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to estimate sample proportions and population proportions.

Math Talks build fluency by encouraging students to think about expressions and rely on what they know about properties of operations to mentally solve a problem.

### Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

### Student Facing

Mentally evaluate the proportion of chips that are blue.

17 are blue out of 50 chips

28 are blue out of 50 chips

17 are blue out of 20 chips

21 are blue out of 60 chips

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate $$\underline{\hspace{.5in}}$$’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to $$\underline{\hspace{.5in}}$$’s strategy?”
• “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . ." or "I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

## 10.2: Pass or Fail (20 minutes)

### Activity

The mathematical purpose of this activity is to collect samples from a population and to use the mean and standard deviation of the sample proportions to estimate the population proportion and its associated margin of error. Students collect samples by drawing slips of paper from a bag and computing the proportion that fit some criteria. The class aggregates their data to create a dot plot showing all the proportions collected from the class. Using the dot plot, students then estimate the population proportion from which the samples are drawn.

### Launch

Arrange students in groups of two. Create a bag for each pair of students. Each bag has 20 slips of paper with 14 of them marked “Pass” and 6 marked “Fail.” Collect data from students about the proportion of slips drawn that are marked Pass.

Tell students that a manufacturer is worried that their product may not be consistently good enough to pass quality control inspections. They are going to take some random samples of their product and have a quality control expert examine the items to determine if they pass or fail. To move forward with production, they need to have 80% of the items pass the inspection.

Provide access to devices that can run GeoGebra or other statistical technology. While they create their dot plots of the data, compute the mean and standard deviation of their sample proportions.

Engagement: Develop Effort and Persistence. Provide a graphic organizer that focuses on increasing the length of on-task orientation in the face of distractions. For example, include a checklist of the tasks students should complete for each trial (draw 10 slips, count Pass slips, find the proportions) and a table for students to record the proportions for each trial.
Supports accessibility for: Attention; Social-emotional skills

### Student Facing

Your teacher will give you a bag with paper slips inside that are marked as either Pass or Fail. Do not empty the bag to look at all of the contents at once.

1. One partner should hold the bag so that the other partner cannot see inside while they draw a slip of paper. The other partner should draw 10 slips of paper from the bag, one at a time. After the 10 slips are drawn, record the number of slips marked Pass.
2. From the results of the first trial, estimate the proportion of the slips in the bag that are marked Pass.
3. Switch roles with your partner and repeat the process until you have run 5 trials. For each trial, compute the proportion of slips you drew that are marked Pass.

trial 1 2 3 4 5
number of Pass slips
proportion of slips marked Pass
4. Create a dot plot based on the trials from the class that shows the proportion of slips drawn that are marked Pass.
5. From the class dot plot, estimate the proportion of slips marked Pass in the bag. Explain your reasoning.

### Activity Synthesis

The purpose of this discussion is for students to share their estimates of the population proportion and to make an estimate for the margin of error. Ask students to share their estimates of the population proportion. Ask students “What could you compute from the data to help you better describe the distribution of the sample proportions? How would these values help estimate the population proportion?” (You could compute the mean and the standard deviation. The mean would be a good estimate and the standard deviation could give an idea of how good of an estimate that is.)

Tell students the mean of all the sample proportions. Ask students, “How confident are you that your estimate is close to the actual population proportion?” Ask students to give a range of values that would be reasonable for the proportion of slips marked Pass. Ask whether they think the manufacturer is likely to have achieved the 80% pass rate based on the samples.

Display the dot plot

Tell students that another group had a dot plot that looked like this.

Here are some questions for discussion:

• “The other group estimated a proportion of around 0.7 of the slips marked pass. Should the other group be more or less confident that their estimate is close to the actual population proportion than we should be about our estimate? Explain your reasoning.” (I think that they should be less confident than us because the data looks more spread out than ours. This means that there should be a bigger interval in which the actual value of the population proportion could fall.)
• “How would the value for the standard deviation of the sample proportions affect your confidence in the answer?” (A greater standard deviation would tell us that the estimate has a wider range of possibilities, so I would be less confident if there was a large standard deviation.)
• Tell students the standard deviation for the class data and, “The standard deviation for the other group in the dot plot is approximately 0.202.” Then ask, “If we want to be 95% confident of the population proportion using the sample proportions, what margin of error should I use for our class data? For the other group’s data?” (The margin of error we would use for each would be two times the standard deviation.)
• “What interval could you be approximately 95% confident contains the population proportion for the class data?” (The interval would be from the mean minus the margin of error to the mean plus the margin of error.)
• “How is this interval related to the margin of error and the standard deviation?” (It is related to both because the margin of error is two times the standard deviation and the interval is centered around the mean.)
Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: “I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.
Design Principle(s): Support sense-making

## 10.3: Fly Memory (10 minutes)

### Activity

The mathematical purpose of this activity is for students to calculate a sample proportion, to estimate the margin of error from a dot plot, and to estimate the margin of error from the mean and standard deviation of the sample proportions resulting from a simulation. In addition, students interpret the meaning of the margin of error in context.

After making an estimation for the population proportion, students are presented with a method for simulating additional samples based on the original sample proportion. Students then use the results from a simulation to measure some expected variability which is used to describe a margin of error.

### Launch

Arrange students in groups of 2. Quiet work time followed by sharing work with a partner.

Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing

### Student Facing

A biologist is breeding fruit flies to include a specific genetic mutation that will be useful in understanding memory in humans. To check whether a fly has the mutation, a DNA sequence is analyzed in a way that kills the fly, so the biologist only wants to test a sample of the flies to estimate the proportion of flies that have the mutation.

The biologist selects 40 flies to sequence at random and finds that 9 of them have the genetic mutation.

1. Based on this sample, estimate the proportion of flies in this group that has the genetic mutation.
2. The scientist is worried that only having one sample may not be reliable for estimating the proportion of flies with the mutation, but does not want to sacrifice more flies to get a larger sample. The proportion from the sample is a good estimate for the population proportion, but it is difficult to understand the possible variability from a single value. Andre has a suggestion for how to better understand the variability:
1. Assume the sample is representative of the population of flies and create a simulation that mimics what the scientist found. Andre gets 200 pieces of paper and marks 45 of them as Mutant and puts them all in a bag. Since Andre decided to use 200 pieces of paper, why should 45 of them be marked Mutant? What are some other combinations of total number of pieces of paper and number marked Mutant that he could use?
2. Andre then simulates the scientist’s sample by drawing a slip of paper from the bag noting whether it is Mutant or not, then replacing the paper into the bag and drawing another paper until he has a sample of 40. He repeats this process for 50 trials and creates a dot plot showing the proportion that are Mutant from each trial. Estimate values on the dot plot a range of proportions that include about 95% of the proportions from the trials.
3. Andre then finds the mean proportion from his simulations to be 0.2195 and the standard deviation to be 0.06. How far are your values from the last question from the mean? This will represent your estimated margin of error.
4. Divide the distance from the last question by the standard deviation to get the margin of error in terms of the number of standard deviations.
3. Based on Andre’s simulations, should the scientist feel confident that the proportion of flies is within two standard deviations of the mean for the simulations?

### Student Facing

#### Are you ready for more?

Suppose the biologist breeds 600 flies. What is the minimum number of flies the biologist should expect to have the mutation based on Andre’s margin of error? What is the maximum that should be expected?

### Anticipated Misconceptions

Students may be confused about why drawing additional samples might have different proportions when they are drawn from a population that is specifically set up to have the same proportion as the original sample. Remind students that different samples from the same population can still have different statistics and ask students where they may have seen that before (such as in the warm-up for this lesson).

### Activity Synthesis

The goal of this discussion is to make sure that students can calculate proportions and estimate the margin of error when given the mean and standard deviation of the sample proportions.

Here are some questions for discussion:

• “How do you think Andre knew to use 200 pieces of paper with 45 of them marked as Mutant?” (He knew that 9 out of 40 is an equivalent proportion to 45 out of 200.)
• "What is the shape of the distribution produced by the simulations?" (It is approximately normal.)
• "Since the distribution is approximately normal, we might be able to use a normal curve to model the data. What proportion of data is within 2 standard deviations of the mean for a normal distribution?" (Close to 95%)
• “What is a different way that you could simulate this experiment?” (You could change the number of slips to be something else is that is an equivalent proportion to 9 out of 40. You could also design a spinner that has a sector that is 81 degrees marked Mutant since that is $$\frac{9}{40}$$ of a circle.)
• “What does the margin of error tell you in this context?” (The margin of error allows you to find the interval that you are 95% confident contains the population proportion.)
• “How confident would you be if you used the standard deviation instead of the margin of error to estimate the interval that contains the mean? Explain your reasoning.” (I would be much less confident. In a previous lesson we noticed that approximately 68% of the data is within one standard deviation of the mean for normally distributed data.)

## Lesson Synthesis

### Lesson Synthesis

Here are some questions for discussion:

• “What does it mean to be 95% confident that the population proportion is within the margin of error of the estimated proportion?” (It means that there is a 95% chance that the actual population proportion falls within the interval.)
• “What does it mean when you say that the population proportion is 0.45 with a margin of error of 0.2?” (It means that there is a 95% chance that the actual population proportion is between 0.25 and 0.65.)
• “Have you ever seen the concept of margin of error in another class, the news, or somewhere else?” (Yes, I notice that they report a margin of error when they do election polls like a person is winning in the polls with 53% of the vote with a margin of error of 3%.)
• “What questions do you have about the margin of error?” (I wonder why we don’t just use the standard deviation. What is the purpose of the margin of error?)

## 10.4: Cool-down - Planet Searching (5 minutes)

### Cool-Down

The next day, their sample has 15 boards that are not straight out of the 50 checked. The manufacturer should estimate that the proportion of boards that are not straight for that day is about 0.3 (since $$\frac{15}{50} = 0.3$$ ) with a margin of error of 0.04 (since twice the standard deviation is $$2 \boldcdot 0.02 = 0.04$$). That means that the actual proportion of boards that are not straight that are produced that day is most likely between 0.26 and 0.34.