NaMe: Unit 4 • Descriptive statistics Lesson 3 ...windrivermath.wikispaces.com/file/view/M1+Unit+4+Lesson+3.pdfUnit 4 • Descriptive statistics Lesson 3: Interpreting Linear Models

Unit 4 • Descriptive statisticsLesson 3: Interpreting Linear Models

NaMe:

Assessment

CCSS IP Math I Teacher Resource© Walch EducationU4-173

Pre-AssessmentCircle the letter of the best answer.

1. Sam tracks the growth of a plant, and records its height in centimeters each week. He determines that the equation y = 2.3x + 16 can be used to estimate the plant’s height for any week. Which statement is true based on Sam’s equation?

a. The plant grows approximately 16 centimeters each week.

b. The starting height of the plant is approximately 16 centimeters.

c. The starting height of the plant is approximately 2.3 centimeters.

d. The plant did not grow during the time Sam tracked its height.

2. Isabella makes deposits to her savings account each month, and she also earns interest. She records the amount of money in her savings account each month, and finds that the equation y = 218x + 100 can be used to estimate the dollars in her savings account for any month. Which statement is true based on Isabella’s equation?

a. She started her account with approximately $218.

b. The amount of money in her account increases by approximately $100 each month.

c. The amount of money in her account increases by approximately $218 each month.

d. Isabella takes approximately $218 out of her account each month.

continued


NaMe:

Assessment

CCSS IP Math I Teacher ResourceU4-174

© Walch Education

3. What is the correlation coefficient, r, of the data in the table below? Use technology to calculate r.

x y3 351 146 672 272 152 138 357 764 524 516 572 18

a. 0.598

b. 0.773

c. 7.321

d. 38.33

4. A data set has a correlation coefficient of –0.916. Which statement about the data set is true?

a. The data has a strong positive linear correlation.

b. The data has a weak positive linear correlation.

c. The data has a weak negative linear correlation.

d. The data has a strong negative linear correlation.

5. Event x and event y have a strong negative linear correlation. Which statement do you know is true about events x and y?

a. If x increases, y decreases.

b. If x increases, y increases.

c. If x increases, it is unknown how y will change.

d. Event x is responsible for the change in y.


Lesson 3: Interpreting Linear ModelsUnit 4 • Descriptive statistics

InstructionCommon Core State Standards

S–ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.★

S–ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.★

S–ID.9 Distinguish between correlation and causation.★

Essential Questions

1. How can you use units to help interpret the slope and y-intercept of a line fitted to data?

2. What is the relationship between the correlation coefficient and the correlation of two events?

3. What is the difference between correlation and causation?

WORDS TO KNOW

causation a relationship between two events where a change in one event is responsible for a change in the second event

correlation a relationship between two events, where a change in one event is related to a change in the second event. A correlation between two events does not imply that the first event is responsible for the change in the second event; the correlation only shows how likely it is that a change also took place in the second event.

correlation coefficient a quantity that assesses the strength of a linear relationship between two variables, ranging from –1 to 1; a correlation coefficient of –1 indicates a strong negative correlation, a correlation coefficient of 1 indicates a strong positive correlation, and a correlation coefficient of 0 indicates a very weak or no linear correlation

linear fit (or linear model) an approximation of data using a linear function

slope the measure of the rate of change of one variable with respect

to another variable; y y

x x

y

xslope

rise

run2 1

2 1

=−−

= =DD

; the slope in the

equation y = mx + b is m.

y-intercept the point at which the graph crosses the y-axis; written as (0, y); the y-intercept in the equation y = mx + b is b.


Instruction


© Walch Education

Recommended Resources• NCTM Illuminations. “Linear Regression I.”

http://walch.com/rr/CAU4L3CorrelationCoefficient

This site allows users to plot points on a coordinate plane. The applet then generates the line of best fit and displays the correlation coefficient.

• Office for Mathematics, Science, and Technology Education, University of Illinois. “Linear Regression Applet.”

http://walch.com/rr/CAU4L3LinearRegression

This applet generates a theoretical line of best fit for points entered by the user. Users then manipulate a slider to change the slope of another line to try to match the theoretical line of best fit. A thermometer shows if the user’s line is a good fit.

• WiseGeek.com. “What is the Difference Between Cause and Correlation?”

http://walch.com/rr/CAU4L3CauseVsCorrelation

This site offers an explanation of the difference between causation and correlation.


NaMe:


Lesson 4.3.1: Interpreting Slope and y-intercept

Warm-Up 4.3.1

A top fuel dragster (a car built for drag racing) can travel 1

4 mile in 4 seconds. The dragster’s

distance over time is graphed below. The graph assumes a constant speed. Use the graph below

to complete problems 1 and 2. Then use what you know about slope-intercept form to answer the

remaining questions.

0 4 8 12 16 20 24 28 32 36 40

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Time in seconds

Dis

tanc

e in

mile

s

1. Find the slope and y-intercept of the function shown in the graph.

2. Write the algebraic equation of the line.

3. What is the slope of a line with the equation y = –x + 7?

4. What is the y-intercept of a line with the equation y = 3x – 2?


Instruction


© Walch Education

Lesson 4.3.1: Interpreting Slope and y-interceptCommon Core State Standard

S–ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.★

Warm-Up 4.3.1 Debrief

A top fuel dragster (a car built for drag racing) can travel 1

4 mile in 4 seconds. The dragster’s

distance over time is graphed below. The graph assumes a constant speed. Use the graph below

to complete problems 1 and 2. Then use what you know about slope-intercept form to answer the

remaining questions.

0 4 8 12 16 20 24 28 32 36 40

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Time in seconds

Dis

tanc

e in

mile

s

1. Find the slope and y-intercept of the function shown in the graph.

The slope is DDy

x or

y

x

change in

change in. To calculate the slope, find any two points on the line.

The graph shows that (0, 0) and (16, 1) are both points on the line. The formula to find the slope between two points (x

1, y

1) and (x

2, y

2) is

y y

x x2 1

2 1

−−

. Substitute (0, 0) and (16, 1) into the formula to find the slope.

y y

x x

16 0

1 0

16

1162 1

2 1

−−

=−

−= =


Instruction


The slope between the two points (0, 0) and (16, 1) is 16.

The y-intercept is the point at which the graph crosses the y-axis. The graph shows that the y-intercept is 0, or (0, 0).

2. Write the algebraic equation of the line.

The equation of a line can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. The equation of the line is y = 16x + 0 or y = 16x.

3. What is the slope of a line with the equation y = –x + 7?

If the equation of a line is in the form y = mx + b, m is the slope of the line.

The slope of the line y = –x + 7 is –1.

4. What is the y-intercept of a line with the equation y = 3x – 2?

If the equation of a line is in the form y = mx + b, then b is the y-intercept.

The y-intercept of y = 3x – 2 is –2, or the point (0, –2).

Connection to the Lesson

• In this lesson, students will need to know how to determine the slope and y-intercept of a linear function using both graphical and algebraic representations.

• This warm-up will remind students how to determine both slope and y-intercept using either representation.

• Students will interpret these values in relation to the real-world model the linear function represents.


Instruction


© Walch Education

Prerequisite Skills

This lesson requires the use of the following skills:

• creating a scatter plot

• finding a linear fit given a scatter plot

• understanding the connection between a graph and an equation of a linear function

• being able to determine the slope and y-intercept of a linear function given either a graph or algebraic equation

IntroductionWhen linear functions are used to model real-world relationships, the slope and y-intercept of the linear function can be interpreted in context. Recall that data in a scatter plot can be approximated using a linear fit, or linear function that models real-world relationships. A linear fit is the approximation of data using a linear function.

The slope of a linear function is the change in the dependent variable divided by the change in

the independent variable, or y

x

change in

change in, sometimes written as

DDy

x. The slope between two points

(x1, y

1) and (x

2, y

2) is

y y

x x2 1

2 1

−−

, and the slope in the equation y = mx + b is m. The slope describes

how much y changes when x changes by 1. When analyzing the slope in the context of a real-world

situation, remember to use the units of x and y in the calculation of the slope. For example, if the

x-axis of a graph represents hours and the y-axis represents miles traveled, the slope of a linear

function graphed on these axes would be change inmiles

change in hours, or the miles traveled each hour.

The y-intercept of a function is the value of y at which the graph of the function crosses the y-axis, or the value of y when x equals 0. When analyzing the y-intercept in a real-world context, this is the starting value of whatever is represented by the y-axis. For example, if the x-axis represents hours and the y-axis represents miles traveled, the y-intercept would be the miles traveled when the number of hours equals 0. The y-intercept in the equation y = mx + b is b. In some cases, the y-intercept doesn’t make sense in context, such as when the quantity of x equals 0, and the y-intercept is something other than 0 (see Example 2).


Instruction


Key Concepts

• The slope of a line with the equation y = mx + b is m.

• The slope of a line is y

x

change in

change in; the slope between two points (x

1, y

1) and (x

2, y

2) is

y y

x x2 1

2 1

−−

.

• In context, the slope describes how much the dependent variable changes each time the independent variable changes by 1 unit.

• The y-intercept of a line with the equation y = mx + b is b.

• The y-intercept is the value of y at which a graph crosses the y-axis.

• In context, the y-intercept is the initial value of the quantity represented by the y-axis, or the quantity of y when the quantity represented by the x-axis equals 0.

Common Errors/Misconceptions

• incorrectly calculating the slope

• confusing the y- and x-intercepts, both in context and when calculating using a graph or equation


Instruction


© Walch Education

Guided Practice 4.3.1Example 1

The graph below contains a linear model that approximates the relationship between the size of a home and how much it costs. The x-axis represents size in square feet, and the y-axis represents cost in dollars. Describe what the slope and the y-intercept of the linear model mean in terms of housing prices.

0 300 600 900 1200 1500 1800 2100 2400 2700 3000

Size in square feet

Cost

in d

olla

rs ($

)

30,000

60,000

90,000

120,000

150,000

180,000

210,000

240,000

270,000

300,000

330,000

360,000

390,000

420,000

450,000

480,000


Instruction


1. Find the equation of the linear fit.

The general equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

Find two points on the line using the graph.

The graph contains the points (300, 60,000) and (600, 120,000).

The formula to find the slope between two points (x1, y

1) and (x

2, y

2)

is y y

x x2 1

2 1

−−

.

Substitute (300, 60,000) and (600, 120,000) into the formula to find the slope.

y y

x x2 1

2 1

−−

Slope formula

120,000 60,000

600 300=

−−

Substitute (300, 60,000) and (600, 120,000) for (x

1, y

1) and (x

2, y

2).

60,000

300200=

Simplify as needed.

The slope between the two points (300, 60,000) and (600, 120,000) is 200.

Find the y-intercept. Use the equation for slope-intercept form, y = mx + b, where b is the y-intercept.

Replace x and y with values from a single point on the line. Let’s use (300, 60,000).

Replace m with the slope, 200. Solve for b.

y = mx + b Equation for slope-intercept form

60,000 = 200(300) + b Substitute values for x, y, and m.

60,000 = 60,000 + b Multiply.

0 = b Subtract 60,000 from both sides.

The y-intercept of the linear model is 0.

The equation of the line is y = 200x.


Instruction


© Walch Education

2. Determine the units of the slope.

Divide the units on the y-axis by the units on the x-axis: dollars

square feet.

The units of the slope are dollars per square foot.

3. Describe what the slope means in context.

The slope is the change in cost of the home for each square foot of the home. The slope describes how price is affected by the size of the home purchased. A positive slope means the quantity represented by the y-axis increases when the quantity represented by the x-axis also increases.

The cost of the home increases by $200 for each square foot.

4. Determine the units of the y-intercept.

The units of the y-intercept are the units of the y-axis: dollars.

5. Describe what the y-intercept means in context.

The y-intercept is the value of the equation when x = 0, or when the size of the home is 0 square feet. For a home with no area, or for no home, the cost is $0.

Example 2

A teller at a bank records the amount of time a customer waits in line and the number of people in line ahead of that customer when he or she entered the line. Describe the relationship between waiting time and the people ahead of a customer when the customer enters a line.

People ahead of customer Minutes waiting1 102 213 325 358 429 45

10 61


Instruction


1. Create a scatter plot of the data.

Let the x-axis represent the number of people ahead of the customer and the y-axis represent the minutes spent waiting.

0 1 2 3 4 5 6 7 8 9 105

10

15

20

25

30

35

40

45

50

55

60

65

Number of people ahead

Min

utes

spe

nt w

aitin

g


Instruction


© Walch Education

2. Find the equation of a linear model to represent the data.

Use two points to estimate a linear model. A line through the two points should have approximately the same number of data values both above and below the line. A line through the first and last data points, (1, 10) and (10, 61), appears to be a good approximation of the data. Find the slope.

The slope between two points (x1, y

1) and (x

2, y

2) is

y y

x x2 1

2 1

−−

. Substitute the points into the formula to find the slope.

y y

x x2 1

2 1

−− Slope formula

61 10

10 1

−−

Substitute (1, 10) and (10, 61) for (x1, y

1)

and (x2, y

2).

51

95.67≈ Simplify as needed.

The slope between the two points (1, 10) and (10, 61) is 5.67≈ .

Find the y-intercept. Use the equation for slope-intercept form, y = mx + b, where b is the y-intercept.

Replace x and y with values from a single point on the line. Let’s use (1, 10).

Replace m with the slope, 5.67. Solve for b.

y = mx + b Equation for slope-intercept form

10 = 1(5.67) + b Substitute values for x, y, and m.

10 = 5.67 + b Simplify.

4.33 = b Subtract 5.67 from both sides.

The y-intercept of the linear model is 4.33.

The equation of the line is y = 5.67x + 4.33.


Instruction



Divide the units on the y-axis by the units on the x-axis: minutes spent waiting

number of people ahead

minutes

person=

The units of the slope are minutes per person.


The slope describes how the waiting time increases for each person in line ahead of the customer. A customer waits approximately 5.67 minutes for each person who is in line ahead of the customer.


The units of the y-intercept are the units of the y-axis: minutes.


The y-intercept is the value of the equation when x = 0, or when the number of people ahead of the customer is 0. The y-intercept is 4.33. In this context, the y-intercept isn’t relevant, because if no one was in line ahead of a customer, the wait time would be 0 minutes. Creating a linear model that matched the data resulted in a y-intercept that wasn’t 0, but this value isn’t related to the context of the situation.


Instruction


© Walch Education

Example 3

For hair that is 12 inches or longer, a hair salon charges for haircuts based on hair length according to the equation y = 5x + 35, where x is the number of inches longer than 12 inches (hair length – 12) and y is the cost in dollars. Describe what the slope and y-intercept mean in context.


Divide the units of the dependent variable, y, by the units of the independent variable, x:

cost in dollars

hair length greater than 12 inches

dollars

inch=


The units of the slope are dollars per inch. The slope describes how the cost of the haircut increases for each inch of hair length greater than 12 inches.


The units of the y-intercept are the units of the dependent variable, y: dollars.


The y-intercept is the value of the equation when x = 0, or when hair length is not greater than 12 inches. The y-intercept is the cost of a haircut when a customer’s hair is no longer than 12 inches. A haircut is $35 if a customer’s hair isn’t longer than 12 inches.


NaMe:


Problem-Based Task 4.3.1: Learning to SpeakDr. Lin is a pediatrician. He tracks how a child’s vocabulary increases when the child first starts speaking. He records the number of months the child has been speaking, and the number of words spoken each month. His data for three different children is in the table below.

Months speaking Words spoken0 50 20 21 321 411 432 822 942 773 963 993 1324 1704 1224 160

One parent whose child was not involved in the study is concerned that her daughter isn’t speaking enough words. When the child had been speaking for 3 months, she spoke 96 words, and now that the child has been speaking for 4 months, she speaks 144 words. What do you think Dr. Lin would say to the concerned parent based on the data he has collected?


NaMe:


© Walch Education

Problem-Based Task 4.3.1: Learning to Speak

Coachinga. Create a scatter plot of the data.

b. Which two data points could be used to find a line to fit the data?

c. What are the units of the slope?

d. Use the units and the problem statement to describe what the slope means in context.

e. What is the rate of change for the vocabulary of the concerned parent’s child?

f. How does the rate of the increase in the child’s vocabulary from part e compare to the slope of the linear model?

g. What do you think Dr. Lin would say to the concerned parent based on the data he has collected?


Instruction


Problem-Based Task 4.3.1: Learning to Speak

Coaching Sample Responsesa. Create a scatter plot of the data.

Let the x-axis represent the number of months the child has been speaking and the y-axis represent the number of words spoken by the child.

0 1 2 3 4

20

40

60

80

100

120

140

160

180

Months speaking

Wor

ds s

poke

n

b. Which two data points could be used to find a line to fit the data?

A line through the two points should have approximately the same number of data values both above and below the line. A line through the points (0, 5) and (2, 77) appears to be a good fit for the data. Draw the line on the scatter plot to see the fit.


Instruction


© Walch Education

0 1 2 3 4

20

40

60

80

100

120

140

160

180

Months speaking

Wor

ds s

poke

n

The general equation of a line in point-slope form is y = mx + b, where m is the slope and b is the

y-intercept. The slope between two points (x1, y

1) and (x

2, y

2) is found using

y y

x x2 1

2 1

−−

, so the slope

between the two points is 77 5

2 036

−−

= .

Use the general equation for slope-intercept form, y = mx + b, to calculate the y-intercept, b. Replace x and y with values from a single point on the line, and replace m with the calculated value. Let’s use (2, 77).

Solve for b.

77 = 36(2) + b Substitute values for x, y, and m.

5 = b Simplify as needed.

The equation of the line is y = 36x + 5.


Instruction


c. What are the units of the slope?

Slope is the change in y divided by the change in x. To find the units of the slope, divide the

units of the dependent variable, y, by the units of the independent variable, x: words

month.

The slope is in words per month.

d. Use the units and the problem statement to describe what the slope means in context.

The slope of the equation describes the change in words spoken each month. This is how many more words a child speaks each month in the first four months after speaking his or her first words. By looking at the slope-intercept form of the equation found in part b, m = 36; thus, each month a child learns approximately 36 words.

e. What is the rate of change for the vocabulary of the concerned parent’s child?

When the child had been speaking for 3 months, she spoke 96 words. After 4 months of

speaking, she spoke 144 words. Substitute (3, 96) and (4, 144) for (x1, y

1) and (x

2, y

2) into the

formula for slope, y y

x x2 1

2 1

−−

.

y y

x x

144 96

4 3442 1

2 1

−−

=−

−=

The rate of change for this child’s vocabulary is 44 words per month.

f. How does the rate of the increase in the child’s vocabulary from part e compare to the slope of the linear model?

The slope in the linear model is 36 words per month. This child’s rate is 44 words per month.

g. What do you think Dr. Lin would say to the concerned parent based on the data he has collected?

Even though the child falls below the linear model, she is still increasing her vocabulary and actually is increasing at a faster rate than the model.

Recommended Closure Activity

Select one or more of the essential questions for a class discussion or as a journal entry prompt.


NaMe:


© Walch Education

Practice 4.3.1: Interpreting Slope and y-interceptA town tracks the number of new homes being built over 10 years. The data is in the table below. Use the table for problems 1–3.

Year New homes1 1302 2333 3404 3405 7096 6427 8098 1,0119 1,324

10 1,511

1. Create a scatter plot of the data set.

2. Find the equation of a line that fits the data.

3. Interpret the slope and y-intercept of the equation in context.

continued


NaMe:


Madeline records the number of homework assignments she has and the total time it takes her to complete her homework. Her data is in the scatter plot below. Use the scatter plot for problems 4 and 5.

0 1 2 3 4 5 6 710

20

30

40

50

60

70

80

90

100

110

120

Homework assignments

Min

utes

to �

nish



continued


NaMe:


© Walch Education

Will practices his basketball free throws. He records the number of free throws he attempts and the number of free throws he makes in the table below. Use the table for problems 6–8.

Free throws attempted Free throws made11 827 2015 911 730 2512 1027 1717 1522 1527 21




continued


NaMe:


A construction company records the number of stories of each building it constructs and the amount of weeks it takes to construct the building. The results are in the scatter plot below. Use the scatter plot for problems 9 and 10.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1610

20

30

40

50

60

70

80

90

100

110

120

130

140

150

Stories

Wee

ks to

com

plet

e




NaMe:


© Walch Education

Lesson 4.3.2: Calculating and Interpreting the Correlation Coefficient

Warm-Up 4.3.2A new social networking company launched a TV commercial. The company tracked the number of users in thousands who joined the network after each week the commercial aired. Use the table of data to answer the questions that follow.

Number of weeks New users in thousands1 62 93 154 195 196 227 328 319 37

10 4511 44


2. Does the data appear to have a linear or exponential relationship? Explain.


Instruction


Lesson 4.3.2: Calculating and Interpreting the Correlation CoefficientCommon Core State Standard

S–ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.★

Warm-Up 4.3.2 DebriefA new social networking company launched a TV commercial. The company tracked the number of users in thousands who joined the network after each week the commercial aired. Use the table of data to answer the questions that follow.

Number of weeks New users in thousands1 62 93 154 195 196 227 328 319 37

10 4511 44


Instruction


© Walch Education


0 1 2 3 4 5 6 7 8 9 10 11 12

5

10

15

20

25

30

35

40

45

50

Number of weeks

New

use

rs in

thou

sand

s

2. Does the data appear to have a linear or exponential relationship? Explain.

The graph of a linear function is a line. The shape of the data in the graph is linear because a line could be drawn on the graph with an approximately equal number of points above and below the graph. The data appears to have a linear relationship.


• In this lesson, students will need to know how to create scatter plots of data sets.

• This warm-up will remind students how to create scatter plots and how to identify whether the relationship between two variables is linear using a graphical representation.

• Students will examine the strength of linear relationships and be introduced to the concept of the correlation coefficient.


Instruction


Prerequisite Skills


• creating a scatter plot given data in a table

• understanding the shape of the graph of a linear function

IntroductionIn previous lessons, we have plotted and analyzed data that appears to have a linear relationship. The data points in some data sets were very close to a linear model, while other data sets had points that were farther from the linear model. The strength of the relationship between data that has a linear trend can be analyzed using the correlation coefficient. A correlation is a relationship between two events, such as x and y, where a change in one event implies a change in another event. The correlation coefficient, r, is a quantity that allows us to determine how strong this relationship is between two events. It is a value that ranges from –1 to 1; a correlation coefficient of –1 indicates a strong negative correlation, a correlation coefficient of 1 indicates a strong positive correlation, and a correlation coefficient of 0 indicates a very weak or no linear correlation. You will use a calculator to calculate the correlation coefficient. Note that a correlation between two events does not imply that changing one event causes a change in the other event—only that a change might have taken place in the other event. This will be explored more later.

Key Concepts

• A correlation is a relationship between two events, where a change in one event implies a change in another event.

• Correlation doesn’t mean that a change in the first event caused a change in the other event.

• The strength of a linear correlation can be measured using a correlation coefficient.

• Before determining the correlation coefficient, look at the scatter plot of the data and make an initial assessment of the strength of a linear relationship between the two events.

• To calculate the correlation coefficient on a graphing calculator, follow the steps on the next page.


Instruction


© Walch Education

On a TI-83/84:

Step 1: Set up the calculator to find correlations. Press [2nd], then [CATALOG] (the “0” key). Scroll down and select DiagnosticOn, then press [ENTER]. (This step only needs to be completed once. The calculator will stay in this mode until changed in this menu.)

Step 2: To calculate the correlation coefficient, first enter the data into a list. Press [2nd], then L1 (the “1” key). Scroll to enter data sets. Press [2nd], then L2 (the “2” key). Enter the second event in L2.

Step 3: Calculate the correlation coefficient. Press [STAT], then select CALC at the top of the screen. Scroll down to 8:LinReg(a+bx), and press [ENTER].

The r value (the correlation coefficient) is displayed along with the equation.

On a TI-Nspire:

Step 1: Go to the lists and spreadsheet page. The icon looks like a table.

Step 2: Enter the data into the first column underneath the shaded row, pressing [enter] after each data value.

Step 3: Use the nav pad to arrow up to the first row below the shaded row and then arrow over to the right so that you are in the second column. Enter the data values, pressing [enter] after each data value.

Step 4: Press the [menu] key.

Step 5: Arrow down to 4: Statistics, and press the center click key.

Step 6: Press the center click key again to select 1: Stat Calculations.

Step 7: Choose 3: Linear Regression (mx+b).

Step 8: At the XList field, press [clear] and then type in “a[]”. To type “[]”, press the [ctrl] key and then the [(] key.

Step 9: Press [tab] to go the YList field and type in “b[]”.

Step 10: Press [tab] to go the Results field and check that results are listed in “c[]”. If not, change them.

Step 11: Press [tab] to “OK” and press the center click key.

Step 12: Arrow down until you see the “r” and look to the right. The number to the right is the correlation coefficient, r.


Instruction


• A correlation coefficient of –1 indicates a strong negative correlation.

• A correlation coefficient of 1 indicates a strong positive correlation.

• A correlation coefficient of 0 indicates a very weak or no linear correlation.

• The correlation coefficient only assesses the strength of a linear relationship between two variables.

• The correlation coefficient does not assess causation—that one event causes the other.


• using the correlation coefficient to analyze data that is not linear

• incorrectly using the correlation coefficient to assess the strength of a relationship


Instruction


© Walch Education


An education research team is interested in determining if there is a relationship between a student’s vocabulary and how frequently the student reads books. The team gives 20 students a 100-question vocabulary test, and asks students to record how many books they read in the past year. The results are in the table below. Is there a linear relationship between the number of books read and test scores? Use the correlation coefficient, r, to explain your answer.

Books read Test score12 238 3

19 149 8

14 5619 1915 256 302 6

14 425 12

15 308 365 191 0

13 634 9

16 7816 167 9


Instruction



Let the x-axis represent books read and the y-axis represent test score.

0 2 4 6 8 10 12 14 16 18 2048121620242832364044485256606468727680

Books read

Test

sco

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2. Describe the relationship between the data using the graphical representation.

It appears that the higher scores were from students who read more books, but the data does not appear to lie on a line. There is not a strong linear relationship between the two events.

3. Calculate the correlation coefficient on your graphing calculator. Refer to the steps in the Key Concepts section.

The correlation coefficient, r, is approximately 0.48.


Instruction


© Walch Education

4. Use the correlation coefficient to describe the strength of the relationship between the data.

A correlation coefficient of 1 indicates a strong positive correlation, and a correlation of 0 indicates no correlation. A correlation coefficient of 0.48 is about halfway between 1 and 0, and indicates that there is a weak positive linear relationship between the number of books a student read in the past year and his or her score on the vocabulary test.

Example 2

A hockey coach wants to determine if players who take many practice shots during practice have a higher shooting percentage. The shooting percentage is calculated by dividing the number of goals scored by the number of shots taken. The coach records the number of practice shots 20 players take each practice, and compares the number with each player’s shooting percentage over the season. Is there a linear relationship between the practice shots and shooting percentage? Use the correlation coefficient, r, to explain your answer.

Practice shots Shooting percentage Practice shots Shooting percentage228 9 223 10164 9 133 764 3 238 10

213 12 228 11166 9 138 860 3 139 7

109 6 118 683 4 210 10

229 13 103 5160 8 114 6


Instruction



Let the x-axis represent the number of practice shots and the y-axis represent the shooting percentage.

0 40 80 120 160 200 2401

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It appears that there is a linear relationship between shots taken per practice and shooting percentage. As the number of practice shots increases, shooting percentage also increases, and the graph appears to have a linear shape.

3. Calculate the correlation coefficient on your graphing calculator. Follow the steps in the Key Concepts section.

The correlation coefficient, r, is 0.94.


Instruction


© Walch Education


A correlation coefficient of 1 indicates a strong positive correlation, and a correlation of 0 indicates no correlation. A correlation coefficient of 0.94 is close to 1, and indicates that there is a strong positive linear relationship between the number of shots taken and the shooting percentage.

Example 3

Caitlyn thinks that there may be a relationship between class size and student performance on standardized tests. She tracks the average test performance of students from 20 different classes, and notes the number of students in each class in the table below. Is there a linear relationship between class size and average test score? Use the correlation coefficient, r, to explain your answer.

Class size Average student test score Class size Average student test score26 28 32 3336 25 27 3029 27 21 3326 32 28 2719 38 23 4134 32 29 2817 43 37 2314 42 14 3923 37 25 3117 41 33 30


Instruction



Let the x-axis represent the number of students in each class and the y-axis represent the average test score.

0 4 8 12 16 20 24 28 32 36 40 444

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As the class size increases, the average test score decreases. It appears that there is a linear relationship with a negative slope between the two variables.

3. Calculate the correlation coefficient on your graphing calculator. Follow the steps in the Key Concepts section.

The correlation coefficient, r, is approximately –0.84.


Instruction


© Walch Education


A correlation coefficient of –1 indicates a strong negative correlation, and a correlation of 0 indicates no correlation. A correlation coefficient of –0.84 is close to –1, and indicates that there is a strong negative linear relationship between class size and average test score.


NaMe:


Problem-Based Task 4.3.2: Good and Bad CholesterolCholesterol is a substance found in human blood. There are two types of cholesterol: HDL (high-density lipoprotein) and LDL (low-density lipoprotein). HDL is a good type of cholesterol, and LDL is the type of cholesterol that can lead to heart attacks and strokes. The sum of HDL and LDL cholesterols is your total cholesterol: HDL + LDL = total cholesterol. The table below shows the total cholesterol and HDL cholesterol for 20 patients, in milligrams per deciliter (mg/dL). Is there a linear relationship between total cholesterol and HDL cholesterol? Use the correlation coefficient, r, to explain your answer.

Total cholesterol (mg/dL) HDL cholesterol (mg/dL)251 47159 30289 63198 54298 75265 53258 86140 45267 49262 71271 50240 40218 47210 57187 31256 52278 79267 50186 58198 38


NaMe:


© Walch Education

Problem-Based Task 4.3.2: Good and Bad Cholesterol


b. What do you think is the relationship between total cholesterol and HDL cholesterol? Use the shape of the scatter plot to explain.

c. Calculate the correlation coefficient, r, using your graphing calculator.

d. What does the correlation coefficient tell you about the relationship between the two events?


Instruction


Problem-Based Task 4.3.2: Good and Bad Cholesterol


Let the x-axis represent total cholesterol and the y-axis represent HDL cholesterol.

140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300

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Total cholesterol (mg/dL)

HD

L ch

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l (m

g/dL

)

0

b. What do you think is the relationship between total cholesterol and HDL cholesterol? Use the shape of the scatter plot to explain.

It appears that the higher values of HDL cholesterol are from people with higher total cholesterol, but the graph does not have a strong linear shape.

c. Calculate the correlation coefficient, r, using your graphing calculator.

The correlation coefficient, r, is approximately 0.596.


Instruction


© Walch Education

d. What does the correlation coefficient tell you about the relationship between the two events?

A correlation coefficient of 1 indicates a strong positive linear correlation, and a correlation coefficient of 0 indicates no linear correlation. A correlation coefficient of 0.596 tells us that there is a positive linear correlation between the two values, but it is not very strong.




NaMe:


Practice 4.3.2: Calculating and Interpreting the Correlation CoefficientFor each of the following scatter plots, describe the type of linear correlation between the two variables: positive, negative, or no correlation, and identify whether it is strong or weak.

1.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

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0 2 4 6 8 10 12 14 16 18 204

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continued


NaMe:


© Walch Education

Warmer weather can be an inspiration to plant gardens and work on landscaping. A plant nursery thinks there may be a relationship between weather and plant sales. Each day, the nursery records the average temperature in ºF and the number of plants sold in a table. Use the table that follows for problems 5–7.

Average temperature (ºF) Plants sold Average temperature (ºF) Plants sold52 18 69 11978 281 64 5976 101 54 2067 152 50 469 113 57 3375 120 76 26356 25 65 5854 37 76 13377 157 78 275


6. Use your graph to describe the relationship between temperature and plant sales.

7. Find the correlation coefficient, r, of the data. Describe what the correlation coefficient indicates about the relationship between the data.

continued


NaMe:


A cruise ship captain wants to know if there is a relationship between the number of children on the ship and the average attendance at a nightly pool party. The ship counted anyone under age 17 as a child. The results are in the table below. Use the table for problems 8–10.

Number of children Average pool party attendance663 23454 76737 23200 112101 116216 139666 23415 52978 61930 62850 22891 63253 110795 22858 64117 144842 65275 136


9. Use your graph to describe the relationship between the number of children and pool party attendance.

10. Find the correlation coefficient, r, of the data. Describe what the correlation coefficient indicates about the relationship between the data.


NaMe:


© Walch Education

Lesson 4.3.3: Distinguishing Between Correlation and Causation

Warm-Up 4.3.3The owner of a guitar store asks customers how many hours a week they practice. He also tracks how much the customers spend. The relationship is shown in the scatter plot below. Use the scatter plot to answer the question that follows.

0 2 4 6 8 10 12 14 16 18 2020

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Practice hours per week

Purc

hase

am

ount

in d

olla

rs ($

)

1. Is there a linear correlation between x and y? Explain.

continued


NaMe:


The data below represents the number of minutes guitarists practice per day and the number of mistakes they make during a performance. Use the data in the table to complete the remaining problems.

x y4 357 263 376 326 298 289 261 404 333 373 38

10 223 38


3. Find the correlation coefficient, r, of the data. Is there a linear relationship between x and y?


Instruction


© Walch Education

Lesson 4.3.3: Distinguishing Between Correlation and CausationCommon Core State Standard

S–ID.9 Distinguish between correlation and causation.★

Warm-Up 4.3.3 DebriefThe owner of a guitar store asks customers how many hours a week they practice. He also tracks how much the customers spend. The relationship is shown in the scatter plot below. Use the scatter plot to answer the question that follows.

0 2 4 6 8 10 12 14 16 18 2020

40

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Practice hours per week

Purc

hase

am

ount

in d

olla

rs ($

)

1. Is there a linear correlation between x and y? Explain.

The shape of a linear graph is a straight line. The data seems to follow a straight line, and it appears that there is a linear correlation between x and y.


Instruction


The data below represents the number of minutes guitarists practice per day and the number of mistakes they make during a performance. Use the data in the table to complete the remaining problems.

x y4 357 263 376 326 298 289 261 404 333 373 38

10 223 38


0 1 2 3 4 5 6 7 8 9 10 11 124

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32

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40

Minutes spent practicing per day

Mis

take

s m

ade

durin

g a

perf

orm

ance


Instruction


© Walch Education

3. Find the correlation coefficient, r, of the data. Is there a linear relationship between x and y?

Use a graphing calculator to find r.

On a TI-83/84:

Step 1: The calculator must first be set up to find correlations. Press [2nd], then [CATALOG] (the “0” key). Scroll down and select DiagnosticOn, then press [ENTER]. (This step only needs to be completed once; the calculator will stay in this mode until changed in this menu.)


Step 3: Now calculate the correlation coefficient. Press [STAT], then select CALC at the top of the screen. Scroll down to 8:LinReg(a+bx), and press [ENTER].

Step 4: The r value (the correlation coefficient) is displayed along with the equation.

On a TI-Nspire:

Step 1: Go to the lists and spreadsheet page. The icon looks like a table.Step 2: Enter the data into the first column underneath the shaded row, pressing [enter] after

each data value. Step 3: Use the nav pad to arrow up to the first row below the shaded row and then arrow

over to the right so that you are in the second column. Enter the data values, pressing [enter] after each data value.

Step 4: Press the [menu] key.Step 5: Arrow down to 4: Statistics and press the center click key.Step 6: Press the center click key again to select 1: Stat Calculations. Step 7: Choose 3: Linear Regression (mx+b).Step 8: At the X List field, press [clear] and then type in “a[]”. To type “[]”, press the [ctrl] key

and then the [(] key.Step 9: Press [tab] to go the Y List field and type in “b[]”.Step 10: Press [tab] to go the Results field and check that results are listed in “c[]”. If not,

change them.Step 11: Press tab to “OK” and press the center click key. Step 12: Arrow down until you see the “r” and look to the right. The number to the right is the

correlation coefficient, r.

Using a calculator, r = –0.97. The correlation coefficient is very close to –1, so there is a strong negative linear correlation between x and y.


Instruction



• In this lesson, students will examine linear correlations between data.

• This warm-up will remind students how to identify linear correlations graphically and by using the correlation coefficient, r.

• Students will differentiate between linear correlations and causations.


Instruction


© Walch Education

Prerequisite Skills


• creating a scatter plot

• identifying linear correlations graphically

• examining linear correlations using the correlation coefficient, r

IntroductionA correlation between two events simply means that there is a consistent relationship between two events, and that a change in one event implies that the other event will change according to the linear relationship. A correlation does not imply causation, or that a change in one event causes the change in the second event. Because many factors influence changes in events, it is very difficult to prove causation, sometimes referred to as a causal relationship. For example, researchers gathered the data below comparing the hours of television watched per day by viewers and each viewer’s weight in pounds.

0 1 2 3 4 5 6 7 8 9 10 11

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Hours of television watched per day

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unds


Instruction


The correlation coefficient, r, between the two events is 0.994. According to this data, a person’s weight could be estimated based on the hours of television he or she watches each day. This is due to the strong correlation between the two events. Can the researchers conclude that watching television is responsible for increased weight, or that watching television causes a person to gain weight?

The correlation does not give any information about why weight increases when a person watches more television. If the data is accurate, it is likely that there is a factor not included in the data set that is associated with both events. For example, a person who watches more television may spend less time exercising and more time sitting. Or, a person who watches more television may eat less healthy foods than someone who spends less time watching television.

There may be many contributing factors that explain how this relationship works. The important note is that watching television by itself cannot cause weight gain. It is also possible that a correlation appears to exist because of a small sample size or a poorly selected sample. Knowing the difference between correlation and causation is particularly important when reading advertisements or other persuasive materials. Think closely about outside factors that may have influenced both events and led to a strong correlation before believing that one event is responsible for the change in another event.

Key Concepts

• Correlation does not imply causation.

• If a change in one event is responsible for a change in another event, the two events have a causal relationship.

• Outside factors may influence and explain a strong correlation between two events.

• Use a calculator to find the correlation coefficient.

On a TI-83/84:

Step 1: Set up the calculator to find correlations. Press [2nd], then [CATALOG] (the “0” key). Scroll down and select DiagnosticOn, then press [ENTER]. (This step only needs to be completed once. The calculator will stay in this mode until changed in this menu.)


Step 3: Calculate the correlation coefficient. Press [STAT], then select CALC at the top of the screen. Scroll down to 8:LinReg(a+bx), and press [ENTER].

The r value (the correlation coefficient) is displayed along with the equation.


Instruction


© Walch Education

On a TI-Nspire:

Step 1: Go to the lists and spreadsheet page. The icon looks like a table.

Step 2: Enter the data into the first column underneath the shaded row, pressing [enter] after each data value.

Step 3: Use the nav pad to arrow up to the first row below the shaded row and then arrow over to the right so that you are in the second column. Enter the data values, pressing [enter] after each data value.

Step 4: Press the [menu] key.

Step 5: Arrow down to 4: Statistics, and press the center click key.

Step 6: Press the center click key again to select 1: Stat Calculations.

Step 7: Choose 3: Linear Regression (mx+b).

Step 8: At the X List field, press [clear] and then type in “a[]”. To type “[]”, press the [ctrl] key and then the [(] key.

Step 9: Press [tab] to go the Y List field and type in “b[]”.

Step 10: Press [tab] to go the Results field and check that results are listed in “c[]”. If not, change them.

Step 11: Press [tab] to “OK” and press the center click key.

Step 12: Arrow down until you see the “r” and look to the right. The number to the right is the correlation coefficient, r.


• assuming that a strong correlation indicates that one event causes another

• failing to consider outside factors that may influence the strength of the correlation between two events


Instruction



Alex coaches basketball, and wants to know if there is a relationship between height and free throw shooting percentage. Free throw shooting percentage is the number of free throw shots completed divided by the number of free throw shots attempted:

free throw shots completed

free throw shots attempted

He takes some notes on the players in his team, and records his results in the table below. What is the correlation between height and free throw shooting percentage? Alex looks at his data and decides that increased height causes a reduced free throw shooting percentage. Is he correct?

Height in inches Free throw percentage Height in inches Free throw percentage75 28 72 2875 22 76 3367 30 76 2580 6 67 5471 43 79 567 40 67 5576 10 78 2576 25 75 1370 42 71 3072 47 68 2979 24 79 1469 23 78 2576 27


Instruction


© Walch Education


Let the x-axis represent height in inches, and the y-axis represent free throw shooting percentage.

0 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 824

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2. Analyze the scatter plot, and describe any relationship between the two events.

As height increases, free throw shooting percentage decreases. It appears that there is a weak negative linear correlation between the two events.

3. Find the correlation coefficient using a graphing calculator. Follow the steps in the Key Concepts section.

r = –0.727


Instruction


4. Describe the correlation between the two events.

–0.727 is close to –1. There is a negative linear correlation between the events.

5. Consider the causal relationship between the two events. Determine if it is likely that height is responsible for the decrease in free throw shooting percentage.

Even if there is a correlation between height and free throw percentage, it is not likely that height causes a basketball player to have more difficulty making free throw shots. If two equally skilled players were of different heights, would you expect one of them to make fewer free throws based only on his or her height? What about a very tall player who spends more time practicing free throws than a very short player? What if the sample size is too small to gather data that’s true for the larger population? There is most likely not a causal relationship between height and free throw percentage.


Instruction


© Walch Education

Example 2

Mr. Gray’s students are interested in learning how studying can improve test performance. Mr. Gray provides students with practice problems related to particular tests. The class records the number of practice problems completed and the score on that related test in the table below. What is the correlation between the number of practice problems completed and the test score? Is there a causal relationship between the number of practice problems completed and the test score?

Problems completed

Test score, out of 100 points

Problems completed

Test score, out of 100 points

10 56 100 7540 70 110 72

100 83 100 7440 54 120 900 45 130 99

50 58 160 10090 72 0 49

150 97 60 5930 50 0 5560 58 180 9690 74 150 100

110 89 30 6730 59 30 56

130 95 20 5010 46


Instruction



Let the x-axis represent the number of completed practice problems and the y-axis represent the test score.

0 20 40 60 80 100 120 140 160 180

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Completed problems

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As the number of completed practice problems increases, the test score also increases. The shape of the graph is approximately a line, and it appears there is a positive linear correlation between the number of practice problems completed and the test score.

3. Find the correlation coefficient using a graphing calculator. Follow the steps in the Key Concepts section.

r = 0.942


Instruction


© Walch Education

4. Describe the correlation between the two events.

0.942 is close to 1. There is a strong positive linear correlation between the number of practice problems completed and the test score.

5. Consider the causal relationship between the two events. Determine if it is likely that the number of practice problems completed is responsible for increased test scores. Note any other factors that could influence test scores.

There is a strong correlation between the two events. A score on a test is related to a student’s knowledge of the test content, and a student’s ability to use content to solve problems. Completing practice problems allows students to develop skills directly related to test performance, and although there are other factors that are related to test performance, it is likely that there is a causal relationship between the number of practice problems completed and the test score.

Example 3

Nadia is a salesperson at a car dealership. She earns money each time she sells a car. To determine if there is a relationship between the number of hours she works and her income, she records the number of hours worked and the amount of money she earns each day. Her data is in the scatter plot that follows. Is there a causal relationship between the hours Nadia works and her daily income?

0 1 2 3 4 5 6 7 8 9 1040

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($)


Instruction



As the number of hours increase, the daily income also increases, but there is much variation in the increase of income as hours increase. There appears to be a weak linear correlation between hours and income.

2. Consider the causal relationship between the two events. Determine if the number of hours worked is responsible for an increase in income.

Nadia earns money when she sells a car. The more hours she works, the greater the opportunity she has to sell more cars. However, the only way to earn more money is to either sell more cars or to sell cars that are more expensive. She could work for 10 hours, but if she doesn’t sell any cars, or make no effort to sell any cars, then her income will reflect this lack of sales or effort. Working more hours does not cause the increase in income. Selling more cars causes the increase in income. There is likely not a causal relationship between hours worked and income earned.


NaMe:


© Walch Education

Problem-Based Task 4.3.3: Good Cholesterol and ExerciseCholesterol is a substance found in human blood. There are two types of cholesterol: HDL (high-density lipoprotein) and LDL (low-density lipoprotein). HDL is a good type of cholesterol, and LDL is the type of cholesterol that can lead to heart attacks and strokes. The sum of HDL and LDL cholesterols is your total cholesterol: HDL + LDL = total cholesterol. A doctor tested 20 patients’ cholesterol levels. She asked each patient how often he or she exercises. The table below shows each patient’s weekly hours of exercise and HDL cholesterol level, in milligrams per deciliter (mg/dL).

Hours of exercise HDL cholesterol (mg/dL)2 470 301 63

1.5 547.5 756 53

8.5 860.5 453 49

6.5 711 507 402 475 571 314 525 79

3.5 502 580 38

The doctor is trying to understand if exercise has an impact on HDL cholesterol. What is the correlation between hours exercised per week and HDL cholesterol level? Is it likely there is a causal relationship between exercise and HDL cholesterol levels?


NaMe:


Problem-Based Task 4.3.3: Good Cholesterol and Exercise


b. Describe the shape of the graph.

c. Does it appear that there is a relationship between exercise and HDL cholesterol levels?

d. What is r, the correlation coefficient?

e. What does r tell us about the relationship between exercise and HDL cholesterol levels?

f. Is it likely that exercising for more hours each week causes higher levels of HDL cholesterol?


Instruction


© Walch Education

Problem-Based Task 4.3.3: Good Cholesterol and Exercise


Let the x-axis represent the hours of exercise per week, and the y-axis represent the HDL cholesterol level in milligrams per deciliter.

0 1 2 3 4 5 6 7 8 98

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Hours of exercise

HD

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b. Describe the shape of the graph.

As the weekly hours of exercise increase, HDL cholesterol generally increases.

c. Does it appear that there is a relationship between exercise and HDL cholesterol levels?

There may be a positive linear relationship between exercise and HDL cholesterol level, based on the shape of the graph.


Instruction


d. What is r, the correlation coefficient?

Using a calculator, r = 0.672.

e. What does r tell us about the relationship between exercise and HDL cholesterol levels?

A correlation coefficient of 1 indicates a strong positive linear relationship, and a correlation coefficient of 0 indicates no correlation. It appears that there is a positive linear correlation between exercise and HDL cholesterol levels.

f. Is it likely that exercising for more hours each week causes higher levels of HDL cholesterol?

The correlation between exercise and HDL cholesterol isn’t strong, which is a clue that there isn’t a causal relationship between the two events. Given that HDL cholesterol generally increases as exercise increases, exercise may have some effect on HDL cholesterol, but it is difficult to determine given the information collected. Other factors, such as genetics and diet, can also influence HDL levels. These factors are not represented in the graph. People who have healthier diets may also exercise more, which could be responsible for the positive linear trend in the data. It is difficult to identify if there is a causal relationship between exercise and HDL cholesterol level given this data set.




NaMe:


© Walch Education

Practice 4.3.3: Distinguishing Between Correlation and CausationA team of advertisers is trying to measure how effectively the advertising campaigns for several products influence purchases of each product. The team records information about total dollars invested in advertising each product in one large city for one year and the total number sold for each product. The data is in the table below. Use the table for problems 1–4.

Advertising spending ($) per product

Products soldAdvertising spending ($)

per productProducts sold

71,000 55,000 31,000 45,00054,000 125,000 88,000 115,00073,000 85,000 32,000 80,00045,000 35,000 80,000 165,00063,000 150,000 34,000 90,00055,000 150,000 76,000 50,00068,000 70,000 38,000 85,00090,000 110,000 67,000 105,00087,000 40,000 48,000 125,00042,000 105,000 18,000 35,00036,000 65,000 37,000 30,00024,000 95,000 90,000 55,00049,000 55,000 26,000 115,00072,000 90,000 89,000 155,00087,000 160,000


2. Describe the shape of the graph.

3. Find the correlation coefficient, r, and describe what this indicates about the relationship between the amount of advertising dollars spent and the number of products sold.

4. Is it likely that there is a causal relationship between the amount of advertising dollars spent and the number of products sold?

continued


NaMe:


A travel agency collects information about its clients. It records a client’s age and the number of countries visited by that client. The data is in the table below. Use the table for problems 5–8.

Age of client Countries visited Age of client Countries visited77 8 47 650 7 48 726 3 30 441 5 26 479 13 46 636 5 53 757 8 77 1128 4 46 573 9 44 745 5 47 833 4 43 546 5 58 737 5 52 670 10 80 928 3


6. Describe the shape of the graph.

7. Find the correlation coefficient, r, and describe what this indicates about the relationship between age and number of countries visited.

8. Is it likely that there is a causal relationship between a client’s age and the number of countries visited?

continued


NaMe:


© Walch Education

A science class studies the time it takes a certain amount of water to reach a boil. Each student uses the same shape container for the water and places the container the same distance from a burner. Each student heats the water at a different temperature, and records that temperature in degrees Fahrenheit. The students record the number of minutes it takes the water to reach a boil given the temperature. The results are in the scatter plot below. Use the scatter plot for problems 9 and 10.

300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600

1

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3

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5

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7

Temperature (°F)

Min

utes

9. Describe the shape of the graph, and describe any possible correlation between temperature and time.

10. Is it likely that there is a causal relationship between the temperature in degrees Fahrenheit and the time it takes the water to boil?

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NaMe: Unit 4 • Descriptive statistics Lesson 3 ...windrivermath.wikispaces.com/file/view/M1+Unit+4+Lesson+3.pdfUnit 4 • Descriptive statistics Lesson 3: Interpreting Linear Models