## Engage NY Eureka Math 7th Grade Module 5 Lesson 20 Answer Key

### Eureka Math Grade 7 Module 5 Lesson 20 Example Answer Key

Example 2: Estimating Population Proportion

Two hundred middle school students at Roosevelt Middle School responded to several survey questions. A printed copy of the responses the students gave to various questions will be provided by your teacher.

The data are organized in columns and are summarized by the following table:

The last column in the data file is based on the question: Which of the following superpowers would you most like to have? The choices were invisibility, super strength, telepathy, fly, or freeze time.

The class wants to determine the proportion of Roosevelt Middle School students who answered “freeze time” to the last question. You will use a sample of the Roosevelt Middle School population to estimate the proportion of the students who answered “freeze time” to the last question.

A random sample of 20 student responses is needed. You are provided the random number table you used in a previous lesson. A printed list of the 200 Roosevelt Middle School students is also provided. In small groups, complete the following exercise:

a. Select a random sample of 20 student responses from the data file. Explain how you selected the random sample.

Answer:

Generate 20 random numbers between 1 and 200. The random number chosen represents the ID number of the student. Go to that ID number row, and record the outcome as “yes” or “no” in the table regarding the freeze time response.

b. In the table below, list the 20 responses for your sample.

Answer:

Answers will vary. Below is one possible result.

c. Estimate the population proportion of students who responded “freeze time” by calculating the sample proportion of the 20 sampled students who responded “freeze time” to the question.

Answer:

Students’ answers will vary. The sample proportion in the given example is \(\frac{5}{20}\), or 0.25.

d. Combine your sample proportion with other students’ sample proportions, and create a dot plot of the distribution of the sample proportions of students who responded “freeze time” to the question.

Answer:

An example is shown below. The class dot plot may differ somewhat from the one below, but the distribution should center at approximately 0.20. (Provide students this distribution of sample proportions if they were unable to obtain a distribution.)

e. By looking at the dot plot, what is the value of the proportion of the 200 Roosevelt Middle School students who responded “freeze time” to the question?

Answer:

0.20

f. Usually, you will estimate the proportion of Roosevelt Middle School students using just a single sample proportion. How different was your sample proportion from your estimate based on the dot plot of many samples?

Answer:

Students’ answers will vary depending on their sample proportions. For this example, the sample proportion is 0.25, which is slightly greater than the 0.20.

g. Circle your sample proportion on the dot plot. How does your sample proportion compare with the mean of all the sample proportions?

Answer:

The mean of the class distribution will vary from this example. The class distribution should center at approximately 0.20.

h. Calculate the mean of all of the sample proportions. Locate the mean of the sample proportions in your dot plot; mark this position with an X. How does the mean of the sample proportions compare with your sample proportion?

Answer:

Answers will vary based on the samples generated by students.

### Eureka Math Grade 7 Module 5 Lesson 20 Exercise Answer Key

Exercises 1–9

Exercise 1.

The first student reported a sample proportion of 0.15. Interpret this value in terms of the summary of the problem in the example.

Answer:

Three of the 20 students surveyed responded that they were vegetarian.

Exercise 2.

Another student reported a sample proportion of 0. Did this student do something wrong when selecting the sample of middle school students?

Answer:

No. This means that none of the 20 students surveyed said that they were vegetarian.

Exercise 3.

Assume you were part of this seventh-grade class and you got a sample proportion of 0.20 from a random sample of middle school students. Based on this sample proportion, what is your estimate for the proportion of all middle school students who are vegetarians?

Answer:

My estimate is 0.20.

Exercise 4.

Construct a dot plot of the 30 sample proportions.

Answer:

Exercise 5.

Describe the shape of the distribution.

Answer:

Nearly symmetrical or mound shaped centering at approximately 0.15

Exercise 6.

Using the 30 class results listed above, what is your estimate for the proportion of all middle school students who are vegetarians? Explain how you made this estimate.

Answer:

About 0.15. I chose this value because the sample proportions tend to cluster between 0.10 and 0.15 or 0.10 and 0.20.

Exercise 7.

Calculate the mean of the 30 sample proportions. How close is this value to the estimate you made in Exercise 6?

Answer:

The mean of the 30 samples to the nearest thousandth is 0.153. The value is close to my estimate of 0.15, and if calculated to the nearest hundredth, they would be the same. (Most likely, students will say between 0.10 and 0.15.)

Exercise 8.

The proportion of all middle school students who are vegetarians is 0.15. This is the actual proportion for the entire population of middle school students used to select the samples. How the mean of the 30 sample proportions compares with the actual population proportion depends on the students’ samples.

Answer:

In this case, the mean of the 30 sample proportions is very close to the actual population proportion.

Exercise 9.

Do the sample proportions in the dot plot tend to cluster around the value of the population proportion? Are any of the sample proportions far away from 0.15? List the proportions that are far away from 0.15.

Answer:

They cluster around 0.15. The values of 0 and 0.30 are far away from 0.15.

### Eureka Math Grade 7 Module 5 Lesson 20 Problem Set Answer Key

Question 1.

A class of 30 seventh graders wanted to estimate the proportion of middle school students who played a musical instrument. Each seventh grader took a random sample of 25 middle school students and asked each student whether or not he or she played a musical instrument. The following are the sample proportions the seventh graders found in 30 samples.

a. The first student reported a sample proportion of 0.80. What does this value mean in terms of this scenario?

Answer:

A sample proportion of 0.80 means 20 out of 25 answered yes to the survey.

b. Construct a dot plot of the 30 sample proportions.

Answer:

c. Describe the shape of the distribution.

Answer:

Nearly symmetrical. It centers at approximately 0.72.

d. Describe the variability of the distribution.

Answer:

The spread of the distribution is from 0.60 to 0.84.

e. Using the 30 class sample proportions listed on the previous page, what is your estimate for the proportion of all middle school students who played a musical instrument?

Answer:

The mean of the 30 sample proportions is approximately 0.713.

Question 2.

Select another variable or column from the data file that is of interest. Take a random sample of 30 students from the list, and record the response to your variable of interest of each of the 30 students.

a. Based on your random sample, what is your estimate for the proportion of all middle school students?

Answer:

Students’ answers will vary depending on the column chosen.

b. If you selected a second random sample of 30, would you get the same sample proportion for the second random sample that you got for the first random sample? Explain why or why not.

Answer:

No. It is very unlikely that you would get exactly the same result. This is sampling variability—the value of a sample statistic will vary from one sample to another.

### Eureka Math Grade 7 Module 5 Lesson 20 Exit Ticket Answer Key

Thirty seventh graders each took a random sample of 10 middle school students and asked each student whether or not he likes pop music. Then, they calculated the proportion of students who like pop music for each sample. The dot plot below shows the distribution of the sample proportions.

Question 1.

There are three dots above 0.2. What does each dot represent in terms of this scenario?

Answer:

Each dot represents the survey results from one student. 0.2 means two students out of 10 said they like pop music.

Question 2.

Based on the dot plot, do you think the proportion of the middle school students at this school who like pop music is 0.6? Explain why or why not.

Answer:

No. Based on the dot plot, 0.6 is not a likely proportion. The dots cluster at 0.3 to 0.5, and only a few dots were located at 0.6. An estimate of the proportion of students at this school who like pop music would be within the cluster of 0.3 to 0.5.