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Chapter 3 – Scatterplots and Linear Regression Name ___________________________

Lesson Objectives: In this chapter we will learn how to graph and summarize two quantitative variables. We will learn how to create and analyze a scatterplot by hand and on the calculator. We will also calculate LSRL (least squares regression lines) and make a predictions based upon our calculated LSRL. Since the LSRL is a model that is used to describe linear trends, we do not expect it to fit the data perfectly unless the data is perfectly linear. To describe how well the line fits the data we will learn how to calculate and interpret correlation, the coefficient of determination, and residual plots.

Date Topics Objectives: Students will be able to … Assignment

Feb

5

3.1 Explanatory and response

variables

3.1 Displaying relationships:

scatterplots

3.1 Interpreting scatterplots

• Describe why it is important to

investigate relationships between

variables.

• Identify explanatory and response

variables in situations where one variable

helps to explain or influences the other.

• Make a scatterplot to display the

relationship between two quantitative

variables.

• Describe the direction, form, and strength

of the overall pattern of a scatterplot.

• Recognize outliers in a scatterplot.

TPS: Read pg. 121 -129, 130 -131,

134 -135

Watch Video #1 -

Take notes.

Complete pg. 2 of Chapter 3

Packet

Feb

6

3.2 Measuring linear association:

correlation

3.2 Facts about correlation

• Know the basic properties of correlation.

• Calculate and interpret correlation.

• Explain how the correlation r is

influenced by extreme observations.

TPS: Read pg. 140 -142, 143 -145,

146 -147

Watch Video #2 -

Take notes.

Complete pg. 3 of Chapter 3

Packet

Feb

7

3.3 Least-squares regression

3.3 Interpreting a regression line

3.3 Prediction

• Interpret the slope and y intercept of a

least-squares regression line.

• Use the least-squares regression line to

predict y for a given x.

• Explain the dangers of extrapolation.

TPS: Read pg. 149 -156, 157 -165

Watch Video #3 -

Take notes.

Complete pg. 4 -5 of Chapter 3

Packet

Feb

8

3.3 Residuals and the least-

squares regression line

3.3 Calculating the equation of

the least-squares regression

line

• Calculate and interpret residuals.

• Explain the concept of least squares.

• Use technology to find a least-squares

regression line.

• Find the slope and intercept of the least-

squares regression line from the means

and standard deviations of x and y and

their correlation.

TPS: Read pg. 167 -172, 176, 181 -

183

Watch Videos #4 - #8;

Take notes.

Complete pg. 6-22 of Chapter 3

Packet

Chapter 3 Packet Due Feb. 11, 2019. Start Chapter 4 Packet and prepare for

cumulative assessment on Feb. 12

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3.1 – Intro to Scatterplots [VIDEO #1]

1. Purpose: ____________________________________________________________________________ 2. Requirements:

a. Explanatory Variable: ________________________________________

b. Response Variable:_____________________________________

c. If there is no distinction:_____________________________________________________

Interpreting Scatterplots

1. Look For An Overall Pattern.

a. Direction: __________________________________________________________________

b. Form: _____________________________________________________________________

c. Strength: __________________________________________________________________

2. Examine Deviations From The Pattern. (____________)

a. Definition: _________________________________________________________________

3. Look For Patterns Among The Different Categorical Variables Of Interest.

a. Note: _____________________________________________________________________

4. Notice Double Data Values.

a. Remember: ________________________________________________________________

Example 1: Scatterplot of State Average SAT Math score vs. Percent of Graduates Taking the SAT’s

Describe the Scatterplot: The average SAT scores are often used to rank state or local school systems. Is this appropriate? Explain.

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3.1 – More Scatterplots [VIDEO #2]

Ex 2: The Sanchez household is about to install solar panels to reduce the cost of heating their house. In order to know how much the solar panels help; they record their consumption of natural gas before the panels are installed. Gas consumption is higher in cold weather, so the relationship between outside temperature and gas consumption is important.

Average Degree-days and Natural Gas Consumption

Month Degree-Days Gas Usage (100 cu ft)

Month Degree-Days Gas Usage (100 cu ft)

November 24 6.3 July 0 1.2

December 51 10.9 August 1 1.2

January 43 8.9 September 6 2.1

February 33 7.5 October 12 3.1

March 26 5.3 November 30 6.4

April 13 4.0 December 32 7.2

May 4 1.7 January 52 11.0

June 0 1.2 February 30 6.9

Note: One degree-day is accumulated for each degree a day’s average temperature falls below 65 F. Scatterplot: Make this plot by hand, and then verify using your graphing calculator. Be sure to label!

Scatterplot (Before & after the Solar Panels have been installed): Describe this scatterplot:

Have the solar panels caused a reduction in gas usage?

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3.1 – Correlation [VIDEO #3] A scatterplot displays the ___________, __________, and _____________ of the relationship between

two quantitative variables. Linear relationships are very important because…

1. _________________________________________________________

2. _________________________________________________________

The stronger the linear relationship, the more useful the data is for making predictions. Correlation: _____________________________________________________________________

_______________________________________________________________________________

Identify which scatterplot has a higher correlation.

(a stronger linear relationship) Plot A:

The moral of the story is…

1. ______________________________

2. ______________________________________

Plot B:

The symbol for correlation is _____ and the formula for correlation is:

1. Enter the data into your calculator.

2. Look at the scatterplot.

Is the relationship positive or negative? __________

Is the relationship strong, moderately strong, no correlation, moderately weak, or weak? ___________

3. Calculate the LSRL.

Press: [STAT], go over to [CALC]. Choose #8: Linreg a+bx.

After this command, you need to tell the calculator what lists to use and where to put the regression

line. Press: [2nd] [1], [ , ], [2nd] [2], [ , ], [VARS] [Y-VARS] [FUNCTION] [1: Y1].

4. Your home screen should look like this: Linreg (a+bx) L1 , L2 , Y1 Press [ENTER]

5. The values of a and b should be displayed as well as r and r2.

−

−

−=

y

i

s

i

s

yy

s

xx

nr

1

1

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Interpreting Correlation:

1. When there is a positive association between the variables, the value of r will be _______________.

Give an example of two variables that are positively associated.

2. When there is a negative association between the variables, the value of r will be _______________.

Give an example of two variables that are negatively associated.

3. Correlation makes no distinction between __________________ and ________________ variables.

4. Correlation requires that both variables be ___________________.

5. Correlation has no ________________________. If you measured height in inches and weight in pounds,

and then converted the height to feet and weight to ounces, the correlation would not change.

6. Correlation is always a number between _____ and _____.

Value: r= -1 r= -.9 r= -.25 r=0 r = .25 r=.9 r=1

Interpretation:

7. Correlation can only measure the strength of _________ relationships. It cannot determine the strength of

curved relationships.

8. Correlation is greatly affected by outliers, therefore it is ________________.

9. If we were to add or subtract the same number for each x or y value the correlation would ________

__________________. Read Example 3.7 on page 145.

Correlation quiz

1. Correlation measures the ______________ and ______________ of the _____________ relationship

between two ________________ variables.

2. Suppose, on a 2-question quiz, question 1 asked you to identify the explanatory and response variables

and you got them backwards. Then question 2, asked you to calculate the correlation between the two

variables. What grade would you get on the quiz?

3. What is wrong with this statement? The correlation between height and weight for all students is r = 0.99

ft-lbs.

4. What is wrong with this statement? The correlation between the amount of time spent exercising and a

person’s body fat percentage is r = -1.25

5. Estimate the correlation in the following two scatterplots:

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3.2 – Least Squares Regression Line - Basics [VIDEO #4] Review:

Correlation measures the _______________ and _________________ of the linear relationship

between two ________________ variables. The symbol for correlation is: _____

If a scatterplot shows a linear relationship, we would like to summarize the overall pattern by drawing a

line through the scatterplot.

Males’ Heights We could use this linear relationship to predict a male’s height, given their age. Height (ft) Ages (Birth – 18 years)

The line we see in the graph above is called a Regression Line. Regression Line - ______________________________________________________________

_____________________________________________________________________________

o Purpose: _______________________________________________________________

o Requirement: ___________________________________________________________

Heights of Mothers and Fathers:

Draw the regression line and predict how the height of

Paul’s mother, given that you know his father’s height

to be 6 ft (72 in).

The Least Squares Regression Line:

Problem: Different people will draw different lines by eye on the scatterplot, especially if the points were more widely scattered than in the example above.

Solution: We need to draw a regression line that doesn’t depend on our guess as to where the line

should go. o No line will pass exactly through all the points. We want one that is as close as possible. o We will use the line to predict ____ from ____, so we want a line that is as close as possible to

the points in the ______________ direction.

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In the example above, if we predicted Paul’s mother’s height to be 64 inches tall and she turned out to actually be 67 inches tall, how much would our error have been?

To minimize our errors, we want a line that makes the vertical distances of the points in a scatterplot from the line as small as possible.

Another word for errors is _______________. The formula for calculating residuals is:

Least Squares Regression Line (LSRL) of y on x is ____________________________________

_____________________________________________________________________________

Every LSRL passes through the point ( ___ , ___ )

• The equation for the LSRL…remember y = mx + b?

o In stats, we will use ________________________ for linear regression.

▪ _____ = ____________________________

▪ _____ = ____________________________

o _____ = y-intercept

▪ When _________, then our predicted ____ will be _____________________.

o _____ = slope

▪ For every _______________, we predict an ________________________ of

____________ in _____________________.

Ex: The LSRL equation to predict the mother’s height from the father’s height is

�̂� = 57.3 + 0.962𝑥. Determine and interpret the slope and y-intercept values.

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3.2 – Least Squares Regression Line - Computation [VIDEO #5]

Calculating the LSRL from Data (with your calculator!!!):

1. Press the “Y =” button and clear out any equations you may have in there.

2. Enter the data from page 4 of your notes regarding temperatures and gas usage into L1 and L2.

3. Now, press [Stat], go right to “CALC”, select #8: “LinReg(a +bx)”. On the home screen, you will see LinReg(a + bx). Tell your calculator what lists to use and where to put the LSRL equation by typing [2nd] “1” to get L1 , then a comma, then [2nd] “2” to get L2 , then another comma, then press [VARS], go right to “Y-VARS”, select “1: Function”, and press enter to select “1: Y1”.

• Your screen should read: LinReg(a+bx) L1 , L2 , Y1

4. Press [Enter]. You should now see the values of a, b, r, and r2 like →

The LSRL equation is �̂� = 1.089 + 0.189𝑥.

5. Press [Zoom] [9: ZoomStat]. You should see this →

Calculating the LSRL from Summary Data: Suppose we have explanatory and response variables and we know that �̅� = 17.222, �̅� = 161.111, 𝒔𝒙 = 19.696, 𝒔𝒚 = 33.479, and r = 0.997. Even though we don’t know the actual data, we can still construct the

LSRL and use it to make predictions. Predict y when x is 30.

To Find the Equation of the LSRL BY HAND: Given Data: x = explanatory variable, y = response variable

LSRL Equation: �̂� = 𝒂 + 𝒃𝒙 Slope Formula: 𝒃 = 𝒓𝒔𝒚

𝒔𝒙

Intercept Formula: 𝒂 = �̅� − 𝒃�̅�

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3.2 – Residual Plots [VIDEO #6]

Residuals

Unless the data are perfectly linear (r = 1 or r = -1) the LSRL is not going to fit the data perfectly. The vertical errors that we observe are called residuals.

Residual = Observed y – Predicted y = 𝑦 − �̂� Example: Does the age at which a child begins to talk predict a later score on a test of mental ability? A study of the development of young children recorded the age in months at which each of the 21 children spoke their first word and Gesell Adaptive Score, the result of an aptitude test taken much later. The data appear below:

Child Age Score Child Age Score

1 15 95 12 9 96

2 26 71 13 10 83

3 10 83 14 11 84

4 9 91 15 11 102

5 15 102 16 10 100

6 20 87 17 12 105

7 18 93 18 42 57

8 11 100 19 17 121

9 8 104 20 11 86

10 20 94 21 10 100

11 7 113

1. Enter the ages into L1 and the scores into L2. 2. Construct a scatterplot on the calculator.

Verify that your plot looks like the one at the right.

3. Calculate the LSRL and sketch it on your scatterplot. Note the following:

LSRL = _______________________________ , r = _______

4. Calculate the residual for the first child. 5. Calculate the residual for the second child. 6. Calculate the residual for the third child.

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3.2 – Residual Plots [VIDEO #7]

7. Use your calculator to calculate each child’s residual. Fill out the first few in the table below:

L1 L2 L3 = L2 – Y1(L1)

Child Age (𝑥) Observed Score

(𝑦) Residual

𝑦 − �̂�

1 15 95

2 26 71

3 10 83

4 9 91

5 15 102

8. Examine the residuals carefully. This is hard to do by just looking at all of the numbers in the table, so lets

construct a graph! To Make a Residual Plot:

a. Plot the explanatory variable on the x-axis and the residuals on the y-axis.

b. Press [2nd] [Statplot]. Make a scatterplot using L1 and L3.

c. Press [Zoom] [9].

d. Press the [Trace] key and examine the residual values.

Things To Notice:

1. When examining a residual plot, we would like to see uniform

scatter of points above and below the line. 2. Does the residual plot have a curved pattern?

If so, the data _________________________ and should not be fit with a LSRL.

3. Do we see increasing or decreasing spread as x increases? If we

see increasing spread, the prediction of y is _________________ for larger x-values. If we see decreasing spread, the prediction of y is less accurate for ____________ x-values.

4. Look for individual points with large residuals. These points are likely outliers! 5. Look for individual points that are extreme in the x-direction. Such

points may not have large residuals, but may be _______________ ___________________. Influential observations pull the LSRL towards themselves. Child 18 is an influential observation. Without child 18 the LSRL is drastically changed (the dashed line).

6. The residuals always add to ______________ (up to round-off error).

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Linear Regression Exploration (almost a complete wrap-up of the chapter).

You will be exploring TWO relationships:

• 1) Can shoe size predict height?

• 2) Can shoe size predict ACT score?

1) Predict whether you believe there will be a high or low correlation between each pair of variables, and

explain your reasoning for each.

2) Graph the scatterplots (one at a time!) on your calculator to see whether your prediction held true.

3) Predict the correlation value, r, for each scatterplot. Then use your calculator to get the actual r value.

Did you overestimate or underestimate each r?

4) With your calculator, write down the LSRL for each scatterplot and make the line appear with your

scatterplot to see that the LSRL is the best fitting line. Interpret the slope and y-intercept for each case.

5) Use YOUR data with the LSRL to predict your height and ACT score from your shoe size.

6) Calculate your residual in each case. Did the LSRL overestimate or underestimate your height and ACT

score?

7) Using your calculator, create residual plots for each set of data. Comment whether you believe the data

would be best represented by a linear model, and explain your reasoning.

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3.2 – The Coefficient of Determination, r2

If a new candy bar were to hit the market, then what is your best

guess at how many calories it will have using this data??

___________________________________________________________

When looking at the Nutritional Facts on a candy bar, what’s typically

the first thing you look at other than the number of calories? ______

Now, the amount of _____does play a role in the number of calories

contained in a candy bar, but is it the ONLY factor??

___________________________________________________________

So, do you think you could predict the number of calories a candy bar has JUST based on the amount

of fat it contains? ________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

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r2 is also called the _________________________________________________________.

r measures the ____________________ and __________________________ of an association.

r2 measures how well the __________ does at reducing ____________________ (or ____________ ).

How to WRITE THE INTERPRETATION of the value of r2:

“ ___________% of the variation in ________________

is explained by the LSRL using ____________.”

1. What is the correlation if about half of the variation in y is accounted for by the LSRL of y on x?

2. What is the largest possible coefficient of determination?

3. What will the scatterplot look like when it takes the value from #2?

4. If you are calculating a LSRL and you mix up the explanatory and response variables, will you get correlation right or wrong? How about the coefficient of determination? How about the LSRL?

5. After calculating the LSRL of distance jumped using the year of the Olympics in which the jumping distance was achieved, you find r2 to be .8440. Interpret this value.

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Interpreting a Residual Plot & Exploring Outliers & Influential Observations

The starting salary of new employees at a firm versus years of education (beyond 8th grade) is fit with a least squares regression line. The graph of the residuals and some computer output for their regression are as follows:

Regression Analysis: Salary (in $1000) versus Education Residuals versus Education

Response is Salary

a) Is a linear model appropriate for the modeling of these data? Clearly explain your reasoning. b) Interpret the slope and y-intercept of the regression line in context.

c) What is the predicted salary of a new employee with 10 years education beyond 8th grade?

d) What is the actual salary of the new employee with 10 years education beyond 8th grade?

Predictor Coef SE Coef T P

Constant 5.840 3.803 1.54 0.163

Education 3.4862 0.5310 6.57 0.000

S = 5.051 R-Sq = 84.3% R-Sq(adj) = 82.4%

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Interpreting a Residual Plot & Exploring Outliers & Influential Observations

The following scatterplot shows the number of teachers (in thousands) for each of the states plus the District of Columbia plotted against the number of students (in thousands) enrolled in grades K-12 for those states and the District of Columbia. The computer printout under the plot shows the results of fitting a straight line to the data by the method of least squares. a. Find and interpret the correlation coefficient for the relationship between the number of teachers and the

student enrollment based on these data.

b. Provide an interpretation of the slope of this line in context.

c. How will the value of the correlation coefficient change if California is removed from the data set and a new correlation coefficient is calculated? (The data point for California is labeled on the scatterplot.) Explain your reasoning.

d. How will the value of the slope of the least squares regression line change if California is removed from the data and a new least squares regression line is calculated? Explain your answer.

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Cautions about r and LSRL Correlation and regression only describe _____________ relationships. Both correlation and the LSRL are strongly influenced by extreme observations, therefore they are

___________________.

Extrapolation: _____________________________________________________________________

__________________________________________________________________________________

Example: Child’s Growth, Fat Rat

Lurking Variable: _________________________________________

__________________________________________________________

__________________________________________________________

o Careful! The relationship you see between two variables may be

strongly influenced by another variable that is lurking in the background.

o A research study found a strong positive correlation between ice cream

sales and murder rate. The article concluded that if people would only

eat less ice cream then there would be less murders.

Example: The Size of American Farms

The number of people living on American Farms has declined steadily during the 1900’s. Here are data on the farm population (millions of persons) from 1935 to 1980.

Year 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980

Population 32.1 30.5 24.4 23 19.1 15.6 12.4 9.7 8.9 7.2

a. Make a scatterplot on your calculator of these data and find the LSRL of farm population on year. b. According to the regression line, how much did the farm population decline each year on average during

this period? _____________ What percent of the observed variation in farm population is accounted for

by linear change over time? ____________

c. Use the regression equation to predict the number of people living on farms in 1990. ____________ Is this

result reasonable? _____ Why? ______________________________________________

The Ideal Solution: The best way to get good evidence that x causes y is to do an __________________ in which x is

changed and lurking variables are kept under control.

Association Does Not Imply Causation!

An association between an explanatory variable (x) and a response variable (y), even if very strong, is NOT (by itself) good evidence that changes in x CAUSE changes in y.

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Upon completion of the Chapter 3 Packet, you should know:

❑ How to construct a scatterplot by hand and on the calculator.

❑ How to identify an explanatory and response variable.

❑ That scatterplots require that both variables be quantitative.

❑ How to interpret a scatterplot. (Form, direction, strength, outliers, patterns among categorical

variables, repeated data values)

❑ The meaning and properties of correlation, r. (r measures the strength and direction of a linear

association)

❑ The properties of a regression line and the conditions for its use.

❑ How to make a prediction based on the LSRL equation.

❑ How to calculate a LSRL with the formulas for “a” and “b”, and with the use of a calculator.

❑ What the coefficient of determination (r2) means. We say that “___% of the variation in y is explained

by the LSRL of y on x.”

❑ How to calculate residuals.

❑ How to construct and interpret a residual plot.