Name_______________________________________________
AP Stats
3.2 Least-Squares Regression Day 1
A ______________________ line is a line that describes how a response variable y changes as an explanatory variable x
changes. We often us it to predict the value of y for a given value of x.
It has an equation of the form:
In this equation,



𝑦̂ (read “y hat”) is the ____________________ value of the response variable y for a given value of the
explanatory variable x.
b is the ___________, the amount by which y is predicted to change when x increases by 1 unit.
a is the ______________________, the predicted value of y when x = 0.
EXAMPLE 1: 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.
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.
EXAMPLE 2: 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 soda  248.6  2.6310  274.9 ml.
EXAMPLE 3: Should we predict how much soda will be left after 60 seconds of tapping?
___________________________ is the use of a regression line for predication far outside the interval of values of the
explanatory variable x used to obtain the line. Such predictions are often not accurate.
A _____________________ is the difference between an observed value of the response variable and the value
predicted by the regression line.
residual = observed y – predicted y = 𝑦 − 𝑦̂
To the left is a visual of residuals… A random
sample of 16 used Ford F-150 SuperCrew 4 x
4 for sale. Graphed is the number of miles
driven and the price. A good regression line
makes the vertical deviations of the points
from the line as small as possible.
EXAMPLE 4: Find and interpret the residual for the can that was tapped for 4 seconds and had 260 ml of soda remaining.
HOMEWORK: p193 #35, 37, 39, 41, 45