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
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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.