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Chapter 3 Describing Relationships Chapter 3 Describing Relationships

Chapter 3 Describing Relationships - PowerPoint Presentation

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Chapter 3 Describing Relationships - PPT Presentation

Section 32 LeastSquares Regression LeastSquares Regression MAKE predictions using regression lines keeping in mind the dangers of extrapolation CALCULATE and interpret a residual INTERPRET the slope and ID: 750531

line regression data interpret regression line interpret data residual ford 150 squares residuals equation driven price miles model variable intercept sample supercrew

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Slide1

Chapter 3

Describing Relationships

Section 3.2Least-Squares RegressionSlide2

Least-Squares RegressionMAKE predictions using regression lines, keeping in mind the dangers of extrapolation.

CALCULATE and interpret a residual.INTERPRET the slope and y intercept of a regression line.DETERMINE the equation of a least-squares regression line using technology or computer output.CONSTRUCT and INTERPRET residual plots to assess whether a regression model is appropriate.Slide3

Least-Squares RegressionINTERPRET the standard deviation of the residuals and r2

and use these values to assess how well a least-squares regression line models the relationship between two variables.DESCRIBE how the least-squares regression line, standard deviation of the residuals, and r2 are influenced by outliers.FIND the slope and y intercept of the least-squares regression line from the means and standard deviations of x and y and their correlation.Slide4

Regression Lines

Linear (straight-line) relationships between two quantitative variables are common. A

regression line

summarizes the relationship between two variables, but only in a specific setting: when one variable helps explain the other.Slide5

Regression Lines

Linear (straight-line) relationships between two quantitative variables are common. A

regression line

summarizes the relationship between two variables, but only in a specific setting: when one variable helps explain the other.Slide6

Regression Lines

Linear (straight-line) relationships between two quantitative variables are common. A

regression line

summarizes the relationship between two variables, but only in a specific setting: when one variable helps explain the other.

A

regression line

is a line that describes how a response variable

y

changes as an explanatory variable

x

changes. Regression lines are expressed in the form

where

(pronounced “y-hat”) is the predicted value of

y

for a given value of

x

.

 Slide7

Prediction

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Predict the price of a Ford F-150 that has been driven 100,000 miles. Slide8

Prediction

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Predict the price of a Ford F-150 that has been driven 100,000 miles. Slide9

Prediction

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Predict the price of a Ford F-150 that has been driven 100,000 miles. 

 Slide10

Prediction

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Predict the price of a Ford F-150 that has been driven 100,000 miles. 

 

 Slide11

Prediction

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Predict the price of a Ford F-150 that has been driven 100,000 miles. 

 

 

 Slide12

Extrapolation

Can we predict the price of a Ford F-150 with 300,000 miles driven?Slide13

Extrapolation

Can we predict the price of a Ford F-150 with 300,000 miles driven?

 Slide14

Extrapolation

Can we predict the price of a Ford F-150 with 300,000 miles driven?

 

 Slide15

Extrapolation

Can we predict the price of a Ford F-150 with 300,000 miles driven?

 

 

 Slide16

Extrapolation

Can we predict the price of a Ford F-150 with 300,000 miles driven?

 

 

 

Extrapolation

is the use of a regression line for prediction far outside the interval of

x

values used to obtain the line. Such predictions are often not accurate.Slide17

Extrapolation

Can we predict the price of a Ford F-150 with 300,000 miles driven?

 

 

 

Extrapolation

is the use of a regression line for prediction far outside the interval of

x

values used to obtain the line. Such predictions are often not accurate.

CAUTION

:

Don’t make predictions using values of

x

that are much larger or much smaller than those that actually appear in your data.Slide18

ResidualsIn most cases, no line will pass exactly through all the points in a scatterplot. Because we use the line to predict y from x, the prediction errors we make are errors in

y, the vertical direction in the scatterplot. These vertical distances are called residuals (the “leftover” variation in the response variable).Slide19

ResidualsIn most cases, no line will pass exactly through all the points in a scatterplot. Because we use the line to predict y from x, the prediction errors we make are errors in

y, the vertical direction in the scatterplot. These vertical distances are called residuals (the “leftover” variation in the response variable).Slide20

Residuals

A

residual

is the difference between the actual value of

y

and the value of

y

predicted by the regression line.

In most cases, no line will pass exactly through all the points in a scatterplot. Because we use the line to predict

y

from

x,

the prediction errors we make are errors in

y,

the vertical direction in the scatterplot. These vertical distances are called residuals (the “leftover” variation in the response variable).Slide21

Residuals

A

residual

is the difference between the actual value of

y

and the value of

y

predicted by the regression line.

In most cases, no line will pass exactly through all the points in a scatterplot. Because we use the line to predict

y

from

x,

the prediction errors we make are errors in

y,

the vertical direction in the scatterplot. These vertical distances are called residuals (the “leftover” variation in the response variable).

 Slide22

Residuals

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Calculate and interpret the residual for the truck that was driven 70,583 miles. Slide23

Residuals

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Calculate and interpret the residual for the truck that was driven 70,583 miles. 

Find the predicted

price

.Slide24

Residuals

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Calculate and interpret the residual for the truck that was driven 70,583 miles. 

 

Find the predicted

price

.Slide25

Residuals

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Calculate and interpret the residual for the truck that was driven 70,583 miles. 

 

 

Find the predicted

price

.Slide26

Residuals

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Calculate and interpret the residual for the truck that was driven 70,583 miles. 

 

 

 

Find the predicted

price

.Slide27

Residuals

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Calculate and interpret the residual for the truck that was driven 70,583 miles. 

Find the residual.

 

 

 

Find the predicted

price

.Slide28

Residuals

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Calculate and interpret the residual for the truck that was driven 70,583 miles. 

 

Find the residual.

 

 

 

Find the predicted

price

.Slide29

Residuals

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Calculate and interpret the residual for the truck that was driven 70,583 miles. 

 

 

Find the residual.

 

 

 

Find the predicted

price

.Slide30

Residuals

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Calculate and interpret the residual for the truck that was driven 70,583 miles. 

 

 

 

Find the residual.

 

 

 

Find the predicted

price

.Slide31

Residuals

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Calculate and interpret the residual for the truck that was driven 70,583 miles. 

 

 

 

Find the residual.

 

 

 

Find the predicted

price

.

Interpret the residual.Slide32

Residuals

A random sample of 16 used Ford F-150 SuperCrew 4 × 4s was selected from among those listed for sale at autotrader.com. The data are shown in the table. For these data, the regression equation is

. Calculate and interpret the residual for the truck that was driven 70,583 miles. 

 

 

 

Find the residual.

 

 

 

Find the predicted

price

.

The actual price of this truck is

$

4765

less than

the cost predicted by the regression line with

x

= miles driven.

Interpret the residual.Slide33

Interpreting a Regression LineA regression line is a model for the data, much like the density curves of Chapter 2. The y intercept and slope of the regression line describe what this model tells us about the relationship between the response variable

y and the explanatory variable x.Slide34

Interpreting a Regression Line

In the regression equation

:

is the

y

intercept

, the predicted value of

y

when

x

= 0

is the

slope

, the amount by which the predicted value of

y

changes when

x increases by 1 unit A regression line is a model for the data, much like the density curves of Chapter 2. The y intercept and slope of the regression line describe what this model tells us about the relationship between the response variable y and the explanatory variable x.Slide35

Interpreting a Regression Line

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the regression equation is

. Interpret the slope of the regression line.Does the value of the y intercept have meaning in this context? If so, interpret the y intercept. If not, explain why.

 Slide36

Interpreting a Regression Line

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the regression equation is

. Interpret the slope of the regression line.Does the value of the y intercept have meaning in this context? If so, interpret the y intercept. If not, explain why.

 

Interpret the slope.Slide37

Interpreting a Regression Line

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the regression equation is

. Interpret the slope of the regression line.Does the value of the y intercept have meaning in this context? If so, interpret the y intercept. If not, explain why.

 

The

predicted

price of a used Ford F-150 goes down by

$

0.1629

(16.29 cents) for each additional mile that the truck has been driven.

Interpret the slope.Slide38

Interpreting a Regression Line

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the regression equation is

. Interpret the slope of the regression line.Does the value of the y intercept have meaning in this context? If so, interpret the y intercept. If not, explain why.

 

The

predicted

price of a used Ford F-150 goes down by

$

0.1629

(16.29 cents) for each additional mile that the truck has been driven.

Interpret the slope.

Interpret the

y

intercept.Slide39

Interpreting a Regression Line

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the regression equation is

. Interpret the slope of the regression line.Does the value of the y intercept have meaning in this context? If so, interpret the y intercept. If not, explain why.

 

The

predicted

price of a used Ford F-150 goes down by

$

0.1629

(16.29 cents) for each additional mile that the truck has been driven.

Interpret the slope.

The

predicted

price (in dollars) of a used Ford F-150 that has been

driven 0 miles.

(The

y intercept does have meaning in this case, as it is possible to have a number of miles driven near 0 miles.)Interpret the y intercept.Slide40

Interpreting a Regression Line

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the regression equation is

. Interpret the slope of the regression line.Does the value of the y intercept have meaning in this context? If so, interpret the y intercept. If not, explain why.

 

The

predicted

price of a used Ford F-150 goes down by

$

0.1629

(16.29 cents) for each additional mile that the truck has been driven.

Interpret the slope.

The

predicted

price (in dollars) of a used Ford F-150 that has been

driven 0 miles.

(The

y intercept does have meaning in this case, as it is possible to have a number of miles driven near 0 miles.)Interpret the y intercept. CAUTION: When asked to interpret the slope or y intercept, it is veryimportant to include the word predicted (or equivalent) in your response. Otherwise, it might appear that you believe the regression equation provides actual values of y.Slide41

The Least-Squares Regression LineThere are many different lines we could use to model the association in a particular scatterplot. A good regression line makes the residuals as small as possible. The regression line we prefer is the one that minimizes the sum of the squared residuals

.Slide42

The Least-Squares Regression LineThere are many different lines we could use to model the association in a particular scatterplot. A good regression line makes the residuals as small as possible. The regression line we prefer is the one that minimizes the sum of the squared residuals

.Slide43

The Least-Squares Regression LineThere are many different lines we could use to model the association in a particular scatterplot. A good regression line makes the residuals as small as possible. The regression line we prefer is the one that minimizes the sum of the squared residuals

.

The

least-squares regression line

is the line that makes the sum of the squared residuals as small as possible.Slide44

p. 184 and 187 Using your CalculatorWe are going to practice using your calculator to make a scatter plot and residual plot.Slide45

Determining if a Linear Model Is Appropriate:Residual PlotsOne of the first principles of data analysis is to look for an overall pattern

and for striking departures from the pattern. A regression line describes the overall pattern of a linear relationship between an explanatory variable and a response variable. We see departures from this pattern by looking at a residual plot.Slide46

Determining if a Linear Model Is Appropriate:Residual PlotsOne of the first principles of data analysis is to look for an overall pattern

and for striking departures from the pattern. A regression line describes the overall pattern of a linear relationship between an explanatory variable and a response variable. We see departures from this pattern by looking at a residual plot.

A

residual plot

is a scatterplot that displays the residuals on the vertical axis and the explanatory variable on the horizontal axis.Slide47

Determining if a Linear Model Is Appropriate:Residual PlotsOne of the first principles of data analysis is to look for an overall pattern

and for striking departures from the pattern. A regression line describes the overall pattern of a linear relationship between an explanatory variable and a response variable. We see departures from this pattern by looking at a residual plot.

A

residual plot

is a scatterplot that displays the residuals on the vertical axis and the explanatory variable on the horizontal axis.Slide48

Determining if a Linear Model Is Appropriate:Residual Plots

A residual plot magnifies the deviations of the points from the line, making it easier to see unusual observations and patterns. If a regression model is appropriate:The residual plot should show no obvious patterns.The residuals should be relatively small in size.Slide49

Determining if a Linear Model Is Appropriate:Residual Plots

A residual plot magnifies the deviations of the points from the line, making it easier to see unusual observations and patterns. If a regression model is appropriate:

The residual plot should show no obvious patterns.

The residuals should be relatively small in size.Slide50

Determining if a Linear Model Is Appropriate:Residual Plots

Pattern in residuals

Linear model not appropriate

A residual plot magnifies the deviations of the points from the line, making it easier to see unusual observations and patterns. If a regression model is appropriate:

The residual plot should show no obvious patterns.

The residuals should be relatively small in size.Slide51

Determining if a Linear Model Is Appropriate:Residual Plots

How to Interpret a Residual PlotTo determine whether the regression model is appropriate, look at the

residual plot.If there is no leftover curved pattern in the residual plot, the regression model is appropriate.If there is a leftover curved pattern in the residual plot, consider using a regression model with a different form.Slide52

Residual Plots

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the least-squares regression equation is

. For this model, technology produced the following residual plot.Is a linear model appropriate for these data? Explain.

 Slide53

Residual Plots

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the least-squares regression equation is

. For this model, technology produced the following residual plot.Is a linear model appropriate for these data? Explain.

 

Because there is no obvious pattern left over in the residual plot, the linear model is appropriate.Slide54

How Well the Line Fits the Data:The Role of s and r2 in Regression

Start here today.Slide55

How Well the Line Fits the Data:The Role of s and r2 in Regression

To assess how well the line fits all the data, we need to consider the residuals for each observation, not just one. Using these residuals, we can estimate the “typical” prediction error when using the least-squares regression line. Slide56

How Well the Line Fits the Data:The Role of s and r2 in Regression

To assess how well the line fits all the data, we need to consider the residuals for each observation, not just one. Using these residuals, we can estimate the “typical” prediction error when using the least-squares regression line.

The

standard deviation of the residuals

s

measures the size of a typical residual. That is,

s

measures the typical distance between the actual

y

values and the predicted

y

values.Slide57

How Well the Line Fits the Data:The Role of s and r2 in Regression

The standard deviation of the residuals s gives us a numerical estimate of the average size of our prediction errors. There is another numerical quantity that tells us how well the least-squares regression line predicts values of the response y. Slide58

How Well the Line Fits the Data:The Role of s and r2 in Regression

The standard deviation of the residuals s gives us a numerical estimate of the average size of our prediction errors. There is another numerical quantity that tells us how well the least-squares regression line predicts values of the response y.

The

coefficient of determination

r

2

measures the percent reduction in the sum of squared residuals when using the least-squares regression line to make predictions, rather than the mean value of

y

. In other words,

r

2

measures the percent of the variability in the response variable that is accounted for by the least-squares regression line.Slide59

How Well the Line Fits the Data:The Role of s and r2 in Regression

The standard deviation of the residuals s gives us a numerical estimate of the average size of our prediction errors. There is another numerical quantity that tells us how well the least-squares regression line predicts values of the response y.

The

coefficient of determination

r

2

measures the percent reduction in the sum of squared residuals when using the least-squares regression line to make predictions, rather than the mean value of

y

. In other words,

r

2

measures the percent of the variability in the response variable that is accounted for by the least-squares regression line.

r

2

tells us how much better the LSRL does at predicting values of

y than simply guessing the mean y for each value in the dataset. Slide60

How Well the Line Fits the Data:The Role of s and r2 in Regression

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the least-squares regression equation is

. For this model, technology gives s = $5740, and r2 = 0.66.Interpret the value of s.

Interpret the value of

r

2

.

 Slide61

How Well the Line Fits the Data:The Role of s and r2 in Regression

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the least-squares regression equation is

. For this model, technology gives s = $5740, and r2 = 0.66.Interpret the value of s.

Interpret the value of

r

2

.

 

Interpret

s

.Slide62

How Well the Line Fits the Data:The Role of s and r2 in Regression

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the least-squares regression equation is

. For this model, technology gives s = $5740, and r2 = 0.66.Interpret the value of s.

Interpret the value of

r

2

.

 

The actual price of a Ford F-150 is typically about

$

5740 away from the price predicted by the least-squares regression line with

x

= miles driven.

Interpret

s

.Slide63

How Well the Line Fits the Data:The Role of s and r2 in Regression

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the least-squares regression equation is

. For this model, technology gives s = $5740, and r2 = 0.66.Interpret the value of s.

Interpret the value of

r

2

.

 

The actual price of a Ford F-150 is typically about

$

5740 away from the price predicted by the least-squares regression line with

x

= miles driven.

Interpret

s

.

Interpret r2.Slide64

How Well the Line Fits the Data:The Role of s and r2 in Regression

Recall that for a random sample of 16 used Ford F-150 SuperCrew 4 × 4s, the least-squares regression equation is

. For this model, technology gives s = $5740, and r2 = 0.66.Interpret the value of s.

Interpret the value of

r

2

.

 

The actual price of a Ford F-150 is typically about

$

5740 away from the price predicted by the least-squares regression line with

x

= miles driven.

Interpret

s

.

About 66% of the variability in the price of a Ford F-150 is accounted for by the least-squares regression line with x = miles driven.Interpret r2.Slide65

Interpreting Computer Regression OutputA number of statistical software packages produce similar regression output.

Be sure you can locate the slope b1the

y intercept b0the values of s

the value of

r

2Slide66

Interpreting Computer Regression OutputA number of statistical software packages produce similar regression output.

Be sure you can locate the slope b1the

y intercept b0the values of s

the value of

r

2Slide67

Interpreting Computer Regression Output

A number of statistical software packages produce similar regression output. Be sure you can locate the slope

b1the y intercept b0

the values of

s

the value of

r

2Slide68

Calculating the Regression Equation from Summary StatisticsUsing technology is often the most convenient way to find the equation of a

least-squares regression line. It is also possible to calculate the equation of the least-squares regression line using only the means and standard deviations of the two variables and their correlation.Slide69

Calculating the Regression Equation from Summary StatisticsUsing technology is often the most convenient way to find the equation of a

least-squares regression line. It is also possible to calculate the equation of the least-squares regression line using only the means and standard deviations of the two variables and their correlation.

How to Calculate the Least-squares Regression Line Using Summary StatisticsWe have data on an explanatory variable x

and a response variable

y

for

n

individuals. From the data, calculate the means

and

and the standard deviations

s

x

and

s

y

of the two variables and their correlation r. The least-squares regression line is the line

with slope and y intercept  Slide70

Regression to the MeanSlide71

Regression to the Mean

The scatterplot shows height versus foot length and the regression equation

. We have added four more lines to the graph:a vertical line at the mean foot length xa vertical line at x +

s

x

a horizontal line at the mean height

y

a horizontal line at

y

+

s

y

 Slide72

Regression to the Mean

The scatterplot shows height versus foot length and the regression equation

. We have added four more lines to the graph:a vertical line at the mean foot length xa vertical line at x +

s

x

a horizontal line at the mean height

y

a horizontal line at

y

+

s

y

 

For an increase of 1 standard deviation in the value of the explanatory variable

x,

the least-squares regression line predicts an increase of

r standard deviations in the response variable y.Slide73

Regression to the Mean

The scatterplot shows height versus foot length and the regression equation

. We have added four more lines to the graph:a vertical line at the mean foot length xa vertical line at x +

s

x

a horizontal line at the mean height

y

a horizontal line at

y

+

s

y

 

For an increase of 1 standard deviation in the value of the explanatory variable

x,

the least-squares regression line predicts an increase of

r standard deviations in the response variable y.This is called regression to the mean, because the values of y “regress” to their mean.Slide74

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.Slide75

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.CORRELATION AND REGRESSION LINES DESCRIBE ONLY LINEAR RELATIONSHIPSSlide76

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.CORRELATION AND REGRESSION LINES DESCRIBE ONLY LINEAR RELATIONSHIPSSlide77

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.CORRELATION AND REGRESSION LINES DESCRIBE ONLY LINEAR RELATIONSHIPS

r = 0.816

r = 0.816

r = 0.816

r = 0.816Slide78

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.Slide79

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.CORRELATION AND LEAST-SQUARES REGRESSION LINES ARE NOT RESISTANTSlide80

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.CORRELATION AND LEAST-SQUARES REGRESSION LINES ARE NOT RESISTANTSlide81

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.CORRELATION AND LEAST-SQUARES REGRESSION LINES ARE NOT RESISTANTSlide82

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.Slide83

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.ASSOCIATION DOES NOT IMPLY CAUSATIONSlide84

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.ASSOCIATION DOES NOT IMPLY CAUSATION

When we study the relationship between two variables, we often hope to show that changes in the explanatory variable cause changes in the response variable.Slide85

Correlation and Regression WisdomCorrelation and regression are powerful tools for describing the relationshipbetween two variables. When you use these tools, you should be aware of

their limitations.ASSOCIATION DOES NOT IMPLY CAUSATION

When we study the relationship between two variables, we often hope to show that changes in the explanatory variable cause changes in the response variable.

CAUTION

:

A strong association between two variables is not enough to draw conclusions about cause and effect.Slide86

Section SummaryMAKE predictions using regression lines, keeping in mind the dangers of extrapolation.CALCULATE and interpret a residual.

INTERPRET the slope and y intercept of a regression line.DETERMINE the equation of a least-squares regression line using technology or computer output.CONSTRUCT and INTERPRET residual plots to assess whether a regression model is appropriate.Slide87

Section SummaryINTERPRET the standard deviation of the residuals and r2 and use these values to assess how well a least-squares regression line models the relationship between two variables.

DESCRIBE how the least-squares regression line, standard deviation of the residuals, and r2 are influenced by outliers.FIND the slope and y intercept of the least-squares regression line from the means and standard deviations of x and y and their correlation.Slide88

Assignment3.2 p. 205-210 #56-68 EOE (Every Other Even) and 70-78 all(56, 60, 64, 68, 70, 71-78 all and Chapter 3 FRAPPY!)

If you are stuck on any of these, look at the odd before or after and the answer in the back of your book. If you are still not sure text a friend or me for help (before 8pm). Tomorrow we will check homework and review for 3.2 Quiz.