## Engage NY Eureka Math 8th Grade Module 6 Lesson 7 Answer Key

### Eureka Math Grade 8 Module 6 Lesson 7 Exercise Answer Key

Example 1.

In the previous lesson, you learned that scatter plots show trends in bivariate data.

When you look at a scatter plot, you should ask yourself the following questions:

a. Does it look like there is a relationship between the two variables used to make the scatter plot?

b. If there is a relationship, does it appear to be linear?

c. If the relationship appears to be linear, is the relationship a positive linear relationship or a negative linear relationship?

To answer the first question, look for patterns in the scatter plot. Does there appear to be a general pattern to the points in the scatter plot, or do the points look as if they are scattered at random? If you see a pattern, you can answer the second question by thinking about whether the pattern would be well described by a line. Answering the third question requires you to distinguish between a positive linear relationship and a negative linear relationship. A positive linear relationship is one that is described by a line with a positive slope. A negative linear relationship is one that is described by a line with a negative slope.

Exercises 1–9

Take a look at the following five scatter plots. Answer the three questions in Example 1 for each scatter plot.

Exercise 1.

Scatter Plot 1

Is there a relationship?

Answer:

Yes

If there is a relationship, does it appear to be linear?

Answer:

Yes

If the relationship appears to be linear, is it a positive or a negative linear relationship?

Answer:

Negative

Exercise 2.

Scatter Plot 2

Is there a relationship?

Answer:

Yes

If there is a relationship, does it appear to be linear?

Answer:

Yes

If the relationship appears to be linear, is it a positive or a negative linear relationship?

Answer:

Positive

Exercise 3.

Scatter Plot 3

Is there a relationship?

Answer:

No

If there is a relationship, does it appear to be linear?

Answer:

Not applicable

If the relationship appears to be linear, is it a positive or a negative linear relationship?

Answer:

Not applicable

Exercise 4.

Scatter Plot 4

Is there a relationship?

Answer:

Yes

If there is a relationship, does it appear to be linear?

Answer:

No

If the relationship appears to be linear, is it a positive or a negative linear relationship?

Answer:

Not applicable

Exercise 5.

Scatter Plot 5

Is there a relationship?

Answer:

Yes

If there is a relationship, does it appear to be linear?

Answer:

Yes

If the relationship appears to be linear, is it a positive or a negative linear relationship?

Answer:

Negative

Exercise 6.

Below is a scatter plot of data on weight in pounds (x) and fuel efficiency in miles per gallon (y) for 13 cars. Using the questions at the beginning of this lesson as a guide, write a few sentences describing any possible relationship between x and y.

Answer:

Possible response: There appears to be a negative linear relationship between fuel efficiency and weight. Students may note that this is a fairly strong negative relationship. The cars with greater weight tend to have lesser fuel efficiency.

Exercise 7.

Below is a scatter plot of data on price in dollars (x) and quality rating (y) for 14 bike helmets. Using the questions at the beginning of this lesson as a guide, write a few sentences describing any possible relationship between x and y.

Answer:

Possible response: There does not appear to be a relationship between quality rating and price. The points in the scatter plot appear to be scattered at random, and there is no apparent pattern in the scatter plot.

Exercise 8.

Below is a scatter plot of data on shell length in millimeters (x) and age in years (y) for 27 lobsters of known age. Using the questions at the beginning of this lesson as a guide, write a few sentences describing any possible relationship between x and y.

Answer:

Possible response: There appears to be a relationship between shell length and age, but the pattern in the scatter plot is curved rather than linear. Age appears to increase as shell length increases, but the increase is not at a constant rate.

Exercise 9.

Below is a scatter plot of data from crocodiles on body mass in pounds (x) and bite force in pounds (y). Using the questions at the beginning of this lesson as a guide, write a few sentences describing any possible relationship between x and y.

Data Source: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0031781#pone-0031781-t001

(Note: Body mass and bite force have been converted to pounds from kilograms and newtons, respectively.)

Answer:

Possible response: There appears to be a positive linear relationship between bite force and body mass. For crocodiles, the greater the body mass, the greater the bite force tends to be. Students may notice that this is a positive relationship but not quite as strong as the relationship noted in Exercise 6.

Example 2: Clusters and Outliers

In addition to looking for a general pattern in a scatter plot, you should also look for other interesting features that might help you understand the relationship between two variables. Two things to watch for are as follows:

CLUSTERS: Usually, the points in a scatter plot form a single cloud of points, but sometimes the points may form two or more distinct clouds of points. These clouds are called clusters. Investigating these clusters may tell you something useful about the data.

OUTLIERS: An outlier is an unusual point in a scatter plot that does not seem to fit the general pattern or that is far away from the other points in the scatter plot.

The scatter plot below was constructed using data from a study of Rocky Mountain elk (“Estimating Elk Weight from Chest Girth,” Wildlife Society Bulletin, 1996). The variables studied were chest girth in centimeters (x) and weight in kilograms (y).

Exercises 10–12

Exercise 10.

Do you notice any point in the scatter plot of elk weight versus chest girth that might be described as an outlier? If so, which one?

Answer:

Possible response: The point in the lower left-hand corner of the plot corresponding to an elk with a chest girth of about 96 cm and a weight of about 100 kg could be described as an outlier. There are no other points in the scatter plot that are near this one.

Exercise 11.

If you identified an outlier in Exercise 10, write a sentence describing how this data observation differs from the others in the data set.

Answer:

Possible response: This point corresponds to an observation for an elk that is much smaller than the other elk in the data set, both in terms of chest girth and weight.

Exercise 12.

Do you notice any clusters in the scatter plot? If so, how would you distinguish between the clusters in terms of chest girth? Can you think of a reason these clusters might have occurred?

Answer:

Possible response: Other than the outlier, there appear to be three clusters of points. One cluster corresponds to elk with chest girths between about 105 cm and 115 cm. A second cluster includes elk with chest girths between about 120 cm and 145 cm. The third cluster includes elk with chest girths above 150 cm. It may be that age and sex play a role. Maybe the cluster with the smaller chest girths includes young elk. The two other clusters might correspond to females and males if there is a difference in size for the two sexes for Rocky Mountain elk. If we had data on age and sex, we could investigate this further.

### Eureka Math Grade 8 Module 6 Lesson 7 Problem Set Answer Key

Question 1.

Suppose data was collected on size in square feet (x) of several houses and price in dollars (y). The data was then used to construct the scatterplot below. Write a few sentences describing the relationship between price and size for these houses. Are there any noticeable clusters or outliers?

Answer:

Answers will vary. Possible response: There appears to be a positive linear relationship between size and price. Price tends to increase as size increases. There appear to be two clusters of houses—one that includes houses that are less than 3,000 square feet in size and another that includes houses that are more than 3,000 square feet

in size.

Question 2.

The scatter plot below was constructed using data on length in inches (x) of several alligators and weight in pounds (y). Write a few sentences describing the relationship between weight and length for these alligators. Are there any noticeable clusters or outliers?

Data Source: Exploring Data, Quantitative Literacy Series, James Landwehr and Ann Watkins, 1987.

Answer:

Answers will vary. Possible response: There appears to be a positive relationship between length and weight, but the relationship is not linear. Weight tends to increase as length increases. There are three observations that stand out as outliers. These correspond to alligators that are much bigger in terms of both length and weight than the other alligators in the sample. Without these three alligators, the relationship between length and weight would look linear. It might be possible to use a line to model the relationship between weight and length for alligators that have lengths of fewer than 100 inches.

Question 3.

Suppose the scatter plot below was constructed using data on age in years (x) of several Honda Civics and price in dollars (y). Write a few sentences describing the relationship between price and age for these cars. Are there any noticeable clusters or outliers?

Answer:

Answers will vary. Possible response: There appears to be a negative linear relationship between price and age. Price tends to decrease as age increases. There is one car that looks like an outlier—the car that is 10 years old. This car has a price that is lower than expected based on the pattern of the other points in the scatter plot.

Question 4.

Samples of students in each of the U.S. states periodically take part in a large-scale assessment called the National Assessment of Educational Progress (NAEP). The table below shows the percent of students in the northeastern states (as defined by the U.S. Census Bureau) who answered Problems 7 and 15 correctly on the 2011 eighth-grade test. The scatter plot shows the percent of eighth-grade students who got Problems 7 and 15 correct on the 2011 NAEP.

a. Why does it appear that there are only eight points in the scatter plot for nine states?

Answer:

Two of the states, New Hampshire and Rhode Island, had exactly the same percent correct on each of the questions, (29,52).

b. What is true of the states represented by the cluster of five points in the lower left corner of the graph?

Answer:

Answers will vary; those states had lower percentages correct than the other three states in the upper right.

c. Which state did the best on these two problems? Explain your reasoning.

Answer:

Answers will vary; some students might argue that Massachusetts at (35,56) did the best. Even though Vermont actually did a bit better on Problem 15, it was lower on Problem 7.

d. Is there a trend in the data? Explain your thinking.

Answer:

Answers will vary; there seems to be a positive linear trend, as a large percent correct on one question suggests a large percent correct on the other, and a low percent on one suggests a low percent on the other.

Question 5.

The plot below shows the mean percent of sunshine during the year and the mean amount of precipitation in inches per year for the states in the United States.

Data source: www.currentresults.com/Weather/US/average-annual-state-sunshine.php

www.currentresults.com/Weather/US/average-annual-state-precipitation.php

a. Where on the graph are the states that have a large amount of precipitation and a small percent of sunshine?

Answer:

Those states will be in the lower right-hand corner of the graph.

b. The state of New York is the point (46,41.8). Describe how the mean amount of precipitation and percent of sunshine in New York compare to the rest of the United States.

Answer:

New York has a little over 40 inches of precipitation per year and is sunny about 45% of the time. It has a smaller percent of sunshine over the year than most states and is about in the middle of the states in terms of the amount of precipitation, which goes from about 10 to 65 inches per year.

c. Write a few sentences describing the relationship between mean amount of precipitation and percent of sunshine.

Answer:

There is a negative relationship, or the more precipitation, the less percent of sun. If you took away the three states at the top left with a large percent of sun and very little precipitation, the trend would not be as pronounced. The relationship is not linear.

Question 6.

At a dinner party, every person shakes hands with every other person present.

a. If three people are in a room and everyone shakes hands with everyone else, how many handshakes take place?

Answer:

Three handshakes

b. Make a table for the number of handshakes in the room for one to six people. You may want to make a diagram or list to help you count the number of handshakes.

Answer:

c. Make a scatter plot of number of people (x) and number of handshakes (y). Explain your thinking.

Answer:

d. Does the trend seem to be linear? Why or why not?

Answer:

The trend is increasing, but it is not linear. As the number of people increases, the number of handshakes also increases. It does not increase at a constant rate.

#### Eureka Math Grade 8 Module 6 Lesson 7 Exit Ticket Answer Key

Question 1.

Which of the following scatter plots shows a negative linear relationship? Explain how you know.

Answer:

Only Scatter Plot 3 shows a negative linear relationship because the y-values tend to decrease as the value of x increases.

Question 2.

The scatter plot below was constructed using data from eighth-grade students on number of hours playing video games per week (x) and number of hours of sleep per night (y). Write a few sentences describing the relationship between sleep time and time spent playing video games for these students. Are there any noticeable clusters or outliers?

Answer:

Answers will vary. Sample response: There appears to be a negative linear relationship between the number of hours per week a student plays video games and the number of hours per night the student sleeps. As video game time increases, the number of hours of sleep tends to decrease. There is one observation that might be considered an outlier—the point corresponding to a student who plays video games 32 hours per week. Other than the outlier, there are two clusters—one corresponding to students who spend very little time playing video games and a second corresponding to students who play video games between about 10 and 25 hours per week.

Question 3.

In a scatter plot, if the values of y tend to increase as the value of x increases, would you say that there is a positive relationship or a negative relationship between x and y? Explain your answer.

Answer:

There is a positive relationship. If the value of y increases as the value of x increases, the points go up on the scatter plot from left to right.