Notes 3.2
Least Squares Regression
Tapping on cans
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.
Tapping on cans
The equation of the regression line in the previous Example is
Problem: Identify the slope and y intercept of the regression line. Interpret
each value in context.
Tapping on cans
For the soda example, the equation of the regression line is soda = 248.6 + 2.63
(tapping time). If we shook a can in the same way as the students did in their
project and tapped on it for 10 seconds, the predicted amount of soda remaining
would be
ml.
Extrapolation and Tapping on Cans
Should we predict how much soda will be left after 60 seconds of tapping? No!
We have data only for cans that were tapped between 0 and 12 seconds. We
don’t know if the linear pattern will continue beyond these values. In fact, if we
did make a prediction for 60 seconds of tapping, we would get 406.4 ml, over
50 ml more than the can originally contained (355 ml)!
Tapping on cans
Problem: Find and interpret the residual for the can that was tapped for 4
seconds and had 260 ml of soda remaining.
Tapping on cans
Here is a scatterplot showing the tapping time and amount of soda remaining for
the 40 cans. The least-squares regression line,
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. The predicted amount
remaining is
The residual is therefore y – = 255 – 269.64 = –14.64 ml. The amount of soda
remaining in this can is 14.64 ml less than expected, based on the tapping time.
Tapping on cans
For the can tapping data, the standard deviation of the residuals is
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.
Tapping on cans
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. The
first scatterplot shows the squared prediction errors when using the mean amount
y as our prediction. When using y as our predicted value, the sum of the squared
prediction errors is 6506.
We could make much better predictions if we knew the tapping time. How much
better? The second scatterplot shows the squared prediction errors when using
the least-squares regression line. The sum of the squared residuals when using
the least-squares regression line is 951.3 (the same quantity we used to calculate
the standard deviation of the residuals, other than a small difference due to
rounding error).
This means that only
= 14.6% of the variation in amount of soda remaining
is unaccounted for by the least-squares regression line. The remaining variation
is due to other factors, such as how vigorously the can was shaken.
Therefore, 1 –
= 85.4% of the variability in amount of soda remaining is
accounted for by the linear model relating amount of soda remaining to tapping
time.
Does seat location affect grades?
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 below.
Problem:
(a) What is the equation of the least-squares 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.
Back to the track!
Here is a scatterplot with the least-squares regression line for predicting the
long-jump distance from sprint time and a scatterplot with the least-squares
regression line for predicting sprint time from long-jump distance.
Does committing more turnovers lead to more points?
In the National Basketball Association, 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.
HW:
pg. 193: 35, 37, 39, 41, 45
HW: pg. 193
43, 47, HW:
49, 52
48, 50, 55, 58
HW:
59, 61, 63, 65, 69, 71-78