# Lesson 12

Estimating a Population Mean

## 12.1: Rolling Distribution (5 minutes)

### Warm-up

The mathematical purpose of this activity is for students to anticipate distributions of values from rolling standard number cubes so that they may compare the values from an actual experiment they will do in the next activity. Ideally, students should anticipate a uniform distribution of values from the number cube, but some variability should not be unexpected.

### Launch

Show students a standard number cube which is a cube with the faces numbered 1 through 6.

### Student Facing

In the next activity, you will roll a standard number cube 35 times.

- Draw a dot plot that shows the distribution of values you might expect for the rolls. Explain your reasoning.
- If you rolled the number cube one million times and found the mean of all the values, what do you expect for the mean? Explain your reasoning.

### Student Response

Student responses to this activity are available at one of our IM Certified Partners

### Activity Synthesis

The purpose of this activity is for students to start thinking about variability and an expected mean after repeated trials. If students do not mention that the distribution should be approximately uniform, bring this up to the class.

Here are some questions for discussion:

- “What do you think the distribution will look like?” (I think it will be relatively uniform but it cannot be perfectly uniform because there is an odd number of rolls and there will be some variability based on the actual rolls.)
- “When you roll the dice 35 times in the next activity, do you think it will look exactly like the dot plot you created?” (I do not think so. There is no way to predict exactly how many of each number you will get on the number cube, but on average it should be uniform.)
- “Do you think the distribution would be more uniform if the die is rolled a million times? Explain your reasoning.” (I think it would be. There might still be some irregularity. However, if you look at it relative to the number of values in each column the difference should be pretty small.)
- “What does this activity make you think about the concept of variation?” (It really makes me wonder about what the standard deviation would be for a million rolls versus 35 rolls. I wonder if I we could pool the class data so that we could look at the standard deviation for several hundred rolls.)

## 12.2: Rolling for Means (15 minutes)

### Activity

The mathematical purpose of this activity is for students to create and analyze a distribution and estimate the mean for rolls of a number cube. Students should recognize the distribution of sample means to be approximately normal which will be important in understanding the margin of error attached to estimates.

### Launch

Arrange students in groups of 4. Collect the data from each student or group to create a dot plot for the class data.

Provide access to devices that can run GeoGebra or other statistical technology.

*Engagement: Develop Effort and Persistence.*Encourage and support opportunities for peer interactions. Students can work with a partner to complete the trials by taking turns rolling the number cube and recording the results.

*Supports accessibility for: Language; Social-emotional skills*

### Student Facing

Roll your number cube applet 35 times, recording the values as you do so.

- Every 5 values, find the mean.
rolls 1 through 5 6 through 10 11 through 15 16 through 20 21 through 25 26 through 30 31 through 35 mean - Share your means with your group and create a dot plot of all the means from your group.
- What do you notice about the shape of the distribution of means?
- Using the dot plot of means, what do you think is a good estimate for the mean of all 140 rolls from your group? How does this value compare to your estimate from the warm-up?

### Student Response

Student responses to this activity are available at one of our IM Certified Partners

### Launch

Arrange students in groups of 4. Distribute one number cube to each student. Collect the data from each student or group to create a dot plot for the class data.

Provide access to devices that can run GeoGebra or other statistical technology.

*Engagement: Develop Effort and Persistence.*Encourage and support opportunities for peer interactions. Students can work with a partner to complete the trials by taking turns rolling the number cube and recording the results.

*Supports accessibility for: Language; Social-emotional skills*

### Student Facing

Roll your number cube 35 times, recording the values as you do so.

- Every 5 values, find the mean.
rolls 1 through 5 6 through 10 11 through 15 16 through 20 21 through 25 26 through 30 31 through 35 mean - Share your means with your group and create a dot plot of all the means from your group.
- What do you notice about the shape of the distribution of means?
- Using the dot plot of means, what do you think is a good estimate for the mean of all 140 rolls from your group? How does this value compare to your estimate from the warm-up?

### Student Response

Student responses to this activity are available at one of our IM Certified Partners

### Anticipated Misconceptions

It can be easy for students to get confused about find the standard deviation and mean of mean values. Encourage students to slow down in their thinking and examine one level at a time. The dot plot consists of mean values and the distribution can be described by its center and spread by using the mean and standard deviation.

### Activity Synthesis

The goal of this discussion is for students to think about how sample means usually produce an approximately normal distribution with some variability. Collect all of the means from the class and create a dot plot. If it does not come up, ask students to describe the distribution shape. It should be approximately normal (bell-shaped and symmetric around 3.5).

Here are some questions for discussion:

- “Why do you think the mean is approximately 3.5?” (The center of the distribution is near 3.5.)
- “Which interval contains most of the means for your group’s data?” (Most of our means were between 2.8 and 4.2.)
- “What percentage of the means are in that interval?” (21 out 28 or approximately 78%)
- “Estimate the percentage of the class data that is in this interval?” (It looks like close to 90% of the means are in this interval.)
- “How does the class distribution compare to your group’s distribution?” (The class distribution is more bell-shaped and more symmetric.)

*Speaking: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.

*Design Principle(s): Support sense-making*

## 12.3: Margin of Error for Means (15 minutes)

### Activity

The mathematical purpose of this activity is for students to practice finding the mean and standard deviation of sample means and to estimate an associated margin of error. Students are presented with several situations, each with several sample means, and asked to estimate the true mean and provide a margin of error.

### Launch

Identify students who talk about the relationship between sample proportions, sample means, and margin of error.

Provide access to devices that can run GeoGebra or other statistical technology.

*Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.*After students read the first situation, present an incorrect statement that represents an incomplete understanding of the “mean of sample means.” For example, “For the 10 gas stations, the mean is $2.52 be because I added them up and divided.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. For example, the author could improve their statement by stating that each number represents the mean from one sample of 25 gas stations. This helps students evaluate, and improve upon, the written mathematical arguments of others, as they reach a deeper understanding of the phrase “mean of sample means.”

*Design Principle(s): Optimize output (for explanation); Maximize meta-awareness*

*Representation: Internalize Comprehension.*Provide appropriate reading accommodations and supports to ensure students have access to the written directions. Some students may benefit from a checklist or list of steps to be able to use the technology.

*Supports accessibility for: Language; Conceptual processing*

### Student Facing

As with the means of sample proportions, the means of sample means are usually within 2 standard deviations of the population mean when there is a large sample size or when the population distribution is approximately normal. For each situation:

- Use the sample means to estimate the mean of the population.
- Find the standard deviation of the sample means.
- Use the standard deviation of the sample means to estimate the margin of error.

- 10 samples of 25 gas stations are selected at random and the price of regular gasoline is recorded for each gas station. The sample means are shown for the 10 samples.
- \$2.38
- \$2.42
- \$2.64
- \$2.35
- \$2.65
- \$2.47
- \$2.67
- \$2.59
- \$2.63
- \$2.41

- The mean number of claimed UFO sightings are shown for 13 samples of 5 randomly selected months.
- 400.2
- 427.4
- 892.2
- 640.6
- 713.4
- 614
- 725.8
- 477.2
- 460
- 445.2
- 476.8
- 336.6
- 536.4

- A company producing baseballs selects 10 baseballs at random 9 times a day and measures the diameter in centimeters. The mean of each of the 9 samples of 10 baseballs is shown.
- 7.5
- 7.6
- 7.2
- 7.4
- 7.2
- 7.3
- 7.5
- 6.9
- 7.5

- A publisher takes 15 random samples of 10 people to determine the number of minutes they spend reading a newspaper. The sample mean is displayed for each of the samples.
- 11.1
- 9.2
- 8.1
- 10.5
- 10
- 9.7
- 7.7
- 11.8
- 11.1
- 7.6
- 6.3
- 9.4
- 10.4
- 8.7
- 10.2

### Student Response

Student responses to this activity are available at one of our IM Certified Partners

### Student Facing

#### Are you ready for more?

Place slips of paper numbered with the integers from 1 to 99 in a paper bag.

- Draw a sample of 10 and record its mean.
- What is the mean absolute deviation of the 10 numbers from their mean?
- Use 50, the actual mean of all of the numbers in the bag, in the calculation of the mean absolute deviation of the 10 numbers you drew. How close is this value to the actual MAD of the sample?

### Student Response

Student responses to this activity are available at one of our IM Certified Partners

### Activity Synthesis

The goal of this discussion is for students to see that the margin of error can be estimated using the standard deviation of the mean of the sample means analogously to using the standard deviation of the mean of the sample proportions. Ask previously identified students, “How is the relationship between sample proportions and margin of error similar to the relationship between sample means and margin of error?” (In both situations you calculate the mean of the sample statistics and find the standard deviation. Once you get the standard deviation you double it to get the margin of error.)

Here are some questions for discussion:

- “What does the margin of error tell you?” (The margin of error tells you that the population mean has a 95% chance of being in the interval from the mean minus the margin of error to the mean plus the margin of error.)
- “What interval is likely to contain the population mean for the UFO question?” (Between 303 and 854 sightings.)
- “Can you think of a situation where sample means would be used to estimate the population mean?” (I think that they could be used to test that products, like boxes of cereal, weigh the correct amount. A box of cereal probably does not weigh exactly what the box says, but probably within a certain margin of error of that amount.)
- “Do you think that increasing the size of a sample would reduce the margin of error? Explain your reasoning.” (I think it would reduce the margin of error because a larger sample should be more representative of the entire sample so it would likely display less variation.)

## Lesson Synthesis

### Lesson Synthesis

Display for all to see. “A company that manufactures tennis balls needs the tennis balls to be between 6.54 centimeters and 6.86 centimeters in diameter in order for them to meet the requirements for use in a tournament.”

Here are some questions for discussion:

- “A company producing tennis balls takes 20 samples of 10 tennis balls and measures the diameter of each of the balls. They report that the mean diameter of balls they produce is 6.59 cm with a margin of error of 0.08 cm. Would you select these tennis balls for use in a tournament? Explain your reasoning” (No, since the actual mean is likely to be between 6.51 cm and 6.67 cm, many of the balls will be too small because they are less than the 6.54 cm requirement for tournaments.)
- “If the estimated mean continues to be 6.59 cm, what is the greatest margin of error that the company could have in order to feel confident that the mean is within the requirements?” (The greatest margin of error is 0.05 cm.)
- “If the mean is 6.59 cm and the margin of error is 0.05 cm then the mean size of tennis balls is likely to be within the requirements for the tournament. What are some ways the company could adjust their manufacturing to ensure that more tennis balls meet the requirements?” (They either need to reduce the margin of error further or increase the mean size of tennis balls so it is closer to the center, 6.7cm, of the interval given by the tournament.)
- “How do you find the margin of error using sample means? Explain your reasoning.” (First, you find several sample means. You then find the standard deviation of the sample means which is a measure of the variability of the sample means. Assuming that the population is approximately normally distributed, you then double the standard deviation to get the margin of error. You double the standard deviation because, in a normal distribution, approximately 95% of the data is within 2 standard deviations of the mean.)

## 12.4: Cool-down - Streaming Games (5 minutes)

### Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

## Student Lesson Summary

### Student Facing

Similar to estimating proportions for populations, we can estimate a population mean based on the mean of several random samples. Using the data from the samples, find the mean of the sample means and the standard deviation of the sample means. The population is very likely to be within 2 sample standard deviations of the mean of the sample means.

For example, a digital clock maker wants to know how well its clocks keep time. They select several random samples of 40 clocks and compare them to an atomic clock to see how many seconds are lost or gained in a day. From each sample, they calculate a sample mean, the mean of the differences between the time, in seconds, on the atomic clock and each digital clock in a given sample.

The mean difference of the sample means is 0.095 seconds (0.095 seconds ahead of the atomic clock time), and the standard deviation of the sample means is 2.791 seconds. The company should expect that the mean difference between the clock time and the actual time for all the clocks it makes is somewhere between -5.487 seconds (\(0.095 - 2 \boldcdot 2.791\)) and 5.677 seconds (\(0.095 + 2 \boldcdot 2.791\)).