Section 3.2
Least – Squares Regression
» summarize the relationship between two
variables, but only in settings where one of the
variables helps explain or predict the other (We
must have an explanatory variable and a
response variable.)
» describe how a response variable y changes as
an explanatory variable x changes
» We often use a regression line to predict the
value of y for a given value of x.
Don’t you hate it when you open a can of soda and some of the contents
spray out of the can? Two AP®Statistics students, Kerry and Danielle,
wanted to investigate if tapping on a can of soda would reduce the
amount of soda expelled after the can has been shaken. For their
experiment, they vigorously shook 40 cans of soda and randomly assigned
each can to be tapped for 0 seconds, 4 seconds, 8 seconds, or 12 seconds.
Then, after opening the can and cleaning up the
mess, the students measured
the amount of soda left in each
can (in ml). Here are the data
and a scatterplot. The scatterplot
shows a fairly strong, positive
linear association between the
amount of tapping time and the
amount remaining in the can. The
line on the plot is a regression line for
predicting the amount remaining
from the amount of tapping time.
A regression lines is a model for the data and provides a
compact mathematical description between the variables.
For the soda example, the equation of the regression line
is 𝑠𝑜𝑑𝑎 = 248.6 + 2.63(𝑡𝑎𝑝𝑝𝑖𝑛𝑔 𝑡𝑖𝑚𝑒).
Identify the slope and y-intercept of the regression line.
Interpret each value in context.
We can use a regression line to predict the response 𝑦
for a specific value of the exploratory variable x. The
accuracy of the prediction depends on how much the
data scatter about the line.
For the soda example, use the equation of the regression
line to predict the amount of soda remaining after
tapping on the can for 10 seconds.
Extrapolation is the use of a regression line for
prediction far outside the interval of values of the
explanatory variable x used to obtain the line. Such
predictions are often not accurate.
Should we predict how much
soda will be left over after
60 seconds of tapping?
Try it! What do you find?
Some data were collected on the weight of a male white
laboratory rate for the first 25 weeks after its birth. A
scatterplot of the weight (in grams) and time since birth (in
weeks) shows a fairly strong, positive linear relationship.
The linear regression equation 𝑤𝑒𝑖𝑔ℎ𝑡 = 100 + 40(𝑡𝑖𝑚𝑒)
models the data fairly well.
1. What is the slope of the regression line?
2. What is the y-intercept? Explain its meaning in context.
3. Predict the rat’s weight after 16 weeks.
4. Should you use this line to predict the rat’s weight at age
2 years?
(There are 454 grams in a pound.)
In most cases, no line will pass exactly through all the
points in a scatterplot. A good regression line makes the
vertical distances of the points from the line as small as
possible.
Recall, the regression equation for “Tapping on Cans”
is 𝑠𝑜𝑑𝑎 = 248.6 + 2.63(𝑡𝑎𝑝𝑝𝑖𝑛𝑔 𝑡𝑖𝑚𝑒).
Find and interpret the residual for the can that was
tapped for 4 seconds and had 260 ml of soda
remaining.
Different regression lines produce different residuals. The
regression line we want is the one that minimizes the sum
of the squared residuals.
The least-squares regression line of y on x is the line that
makes the sum of the squared residuals as small as
possible.
Find the value of each residual.
What do you notice if you add them
together?
Where else have we seen this?
We want the sum of squared of our
residuals to be as small as
possible!!
1. Enter your data into two lists.
2. Create a scatterplot.
Describe what you see. (DOFS)
3. Find the equation of your leastsquares regression line.
STAT -> CALC -> LinReg(a+bx)
4. To store your equation as a line:
VARS -> Y-VARS -> Function: Y1
» Page 163 #27 – 32 (More review from 3-1)
» Page 193 #35, 37, 39, 41, 43, 45, 47
Here is a scatterplot showing the tapping time and amount
of soda remaining for the 40 cans. The least-squares
regression line, 𝑠𝑜𝑑𝑎 = 248.6 + 2.63(𝑡𝑎𝑝𝑝𝑖𝑛𝑔 𝑡𝑖𝑚𝑒), is
shown on the scatterplot. The point in red is for the can
that was tapped for 8 seconds and had 255 ml remaining
after it was opened.
What is the residual for this
point? Explain what the value
means.
A regression line describes the overall pattern of a linear
relationship between two variables. We see departures
from this pattern by looking at the residuals.
The residuals for a least-squares regression line have a
special property: the mean of the least squares residuals is
always zero
A residual plot is a scatterplot of the residuals against the
explanatory variable. Residual plots help us assess how
well a regression line fits the data.
Here are the scatterplot and residual plot for the can
tapping data.
**Notice the horizontal axis is the same.
**The “residual = 0” corresponds to the regression line.
» The residual plot in effect turns the regression line
horizontal and magnifies deviations so they are easier
to see.
» The scatterplot and residual plot show a nonlinear
relationship.
» When an obvious curved pattern exists in a residual
plot, the model we are using is not appropriate.
» When we use a line to model a linear association, there
should only be a random scatter of points. (Consider
the scatterplot and residual plot for price versus miles
driven.)
» Residuals look for what is left over when
comparing the actual observation with the
predicted value.
𝑦 − 𝑦 = 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 − 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑
» A residual plot looks at the form that is left
when comparing the form of the association to
the form of the regression model
˃ If there is a pattern, then the form of the two
plots are not the same.
The data below represents the years since 1995 and
student enrollment. Based on the residual plot, is a
linear model appropriate for these data?
*Type your data into two lists.
*Enter your residuals. [2ndStat]
*Create a scatter plot with
Explanatory variables and residuals
The plots below are the scatterplot and residual plot for
years since 1995 and student enrollment. What do you
notice?
An association can be clearly nonlinear and still have
a correlation close to ±1.
» Residuals show the amount of error for each
observation.
» To estimate the approximate size of a “typical”
prediction error (residual), calculate the standard
deviation of the residuals.
𝑠=
𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠 2
=
𝑛−2
𝑦𝑖 − 𝑦
𝑛−2
2
For the can tapping data, the standard deviation of
the residuals is 𝑠 =
951.536
40−2
= 5.00 𝑚𝐿.
When we use the least-squares regression line to
predict the amount of soda remaining using the
amount of tapping time, our predictions will typically
be off by about 5 ml.
If all of the points fall directly on the least squares line,
the sum of squared residuals is 0, which means 𝑟 2 = 1.
This means all the variation in y is accounted for by the
linear relationship with x.
Suppose that we wandered in during the can tapping
experiment and found a partially-full can. Without
measuring the contents, how could we predict how much
soda is left in the can? We don’t know how long it was
tapped, so our best guess would be the mean amount
remaining in all the cans: 𝑦 = 264.45 mL.
When using 𝑦 as our predicted value, the sum of the
squared prediction errors is 6506.
The sum of the squared residuals when using the leastsquares regression line is 951.3.
“ ______ % of the variation in
(y variable) is accounted for by the
linear relationship to the (x variable).”
» They both help to answer the question, “How
well does the line fit the data?”
» S is measured in the same units as the response
variable (y).
» 𝑟 2 does not have units but is usually expressed
as a percentage.
» Report BOTH when you are assessing the fit of a
line.
In Section 3.1, we looked at the
relationship between the 40-yard sprint
time (in seconds) and the long-jump
distance (in inches) for a small statistics
class with 12 students.
1. Use your graphing calculators to
create a scatterplot and find the
least-squares regression line.
2. Find the value of 𝑟 2 .
3. Using your residuals, create a
residual plot.
4. Find the value of s.
A scatterplot with the least-squares regression line
( 𝑦 = 414.79 – 45.74x) and a residual plot are shown
below. Also, s = 22.38 and 𝑟 2 = 0.702.
(a) Calculate and interpret the residual for Christian, who
had a sprint time of 7.25 seconds and a long jump of
110 inches.
(b) Is a linear model appropriate for these data? Explain.
(c) Interpret the value of s.
(d) Interpret the value of 𝑟 2 .
Section 3-2: Page 193 #48, 49, 50, 51, 55, 58
Many people believe that students learn better if they sit
closer to the front of the classroom. Does sitting closer
cause higher achievement, or do better students simply
choose to sit in the front? To investigate, an AP®Statistics
teacher randomly assigned students to seat locations in
his classroom for a particular chapter. At the end of the
chapter, he recorded the row number (row 1 is closest to
the front) and test score for each student. Least-squares
regression was performed on the data. A scatterplot with
the regression line added, a residual plot, and some
computer output from the regression are shown.
(a) What is the equation of the leastsquares regression line that describes
the relationship between row number
and test score? Define any variables
that you use.
(b) Interpret the slope of the regression
line in context.
(c) Find the correlation.
(d) Is a line an appropriate model to
use for these data? Explain how you
know.
It is also possible to calculate the equation of the leastsquares regression line using only the means and standard
deviations of the two variables and their correlation.
Both formulas are on the AP Test formula sheet!!
In the previous example, we investigated the relationship
between test scores and seat location. The mean and
standard deviation of the row numbers are 𝑥 = 4.033 and
𝑠𝑥 = 1.974. The mean and standard deviation of the test
scores are 𝑦 = 81.2 and s𝑦 = 10.135. The correlation
between row number and test score is r = –0.218. (Note
that this value is slightly different than the previous
example because of rounding in the computer output.)
Find the equation of the least-squares regression line for
predicting test score from row number. Show your work.
Netbooks are a hybrid of a laptop
computer and a tablet. They are
smaller and have better battery life
than a traditional laptop. They also
have a separate keyboard, unlike
most tablets. Consumer Reports did
a study of 22 netbooks in their
February 2010 issue. Among the
variables measured were battery life
(hours), weight (pounds), and cost.
The data appear in the table. Should
we use a linear model to predict
battery life of a new netbook based
on its weight? If so, how accurate
will our predictions be?
1. The distinction between explanatory and response
variables is important in regression.
2. Correlation and regression lines describe only linear
relationships.
3. Correlation and least-squares regression lines are not
resistant.
An outlier is an observation that lies outside the overall
pattern of the other observations. Points that are outliers
in the y direction but not the x direction of a scatterplot
have large residuals. Other outliers may not have large
residuals.
An observation is influential for a statistical calculation if
removing it would markedly change the result of the
calculation. Points that are outliers in the x direction of a
scatterplot are often influential for the least-squares
regression line.
4. Association does not imply causation.
(Use common sense when drawing conclusions.)
Example – Does committing more turnovers
lead to more points?
In the NBA, there is a strong positive association between the
number of turnovers a player has and the number of points
that he scores. A turnover is when a player loses the ball to
the other team. Could a player increase his point totals by
turning the ball over more frequently? No! Turning the ball
over to the other team doesn’t cause a player to score more
points. Instead, there is another variable that influences both
turnovers and points: playing time. Players who are on the
court more often tend to score more points and have more
turnovers than players who don’t get much playing time.
In the chapter-opening Case Study (page 141), the Starnes
family had just missed seeing Old Faithful erupt. They
wondered how long it would be until the next eruption.
The scatterplot below shows data on the duration (in
minutes) and the interval of time until the next eruption
(also in minutes) for each Old Faithful eruption in the
month before the eruption. Answer all of the questions on
page 191 and turn them in.
Section 3-2: Page 193 #59, 61, 63, 65, 69, 71 – 78