AP Statistics Section 3.2 A
Regression Lines
Linear relationships between two
quantitative variables are quite common.
Just as we drew a density curve to model
the data in a histogram, we can summarize
the overall pattern in a linear relationship
by drawing a _______________
regression line on the
scatterplot.
Note that regression requires that
we have an explanatory variable
and a response variable. A
regression line is often used to
predict the value of y for a given
value of x.
A least-squares regression line
relating y to x has an equation of
the form ___________
yˆ  a  bx
In this equation, b is the _____,
slope
and a is the __________.
y-intercept
NOTE: You must always define the
variables (i.e. yˆ and x) in your
regression equation.
The formulas below allow you to find
the value of b depending on the
information given in the problem:
xi  x  yi  y 

b
2
 xi  x 
br
Sy
Sx
Once you have computed b, you
can then find the value of a using
this equation.
a  y  b(x )
TI-83/84: Do the exact same steps
involved in finding the correlation
coefficient, r.
Example 1: Let’s revisit the data
from section 3.1A on sparrowhawk
colonies and find the regression
equation.
# new birds  31.934 - .304(% of birds returning)
Interpreting b: The slope b is the
rate of change in the
predicted _____________
response variable y as the
explanatory variable x increases by
1.
Example 2: Interpret the slope of
the regression equation for the
data on sparrowhawk colonies.
For each increase of 1% in the number of adult birds
returning to the colony the next year, the predicted
number of new birds decreases by .304
You cannot say how important a
relationship is by looking at how
big the regression slope is.
Interpreting a: The y-intercept a is
the value of the response variable
when the explanatory variable is
equal to ____.
0
Example 3: Interpret the yintercept of the regression
equation for the data on
sparrowhawk colonies.
When the percent of birds returning is 0, the predicted
number of new birds in the colony is 31.934.
Example 4: Use your regression equation for the
data on sparrowhawk colonies to predict the
number of new birds coming to the colony if
87% of the birds from the previous year return.
yˆ  31.934  .304(87)
yˆ  5.486
CAUTION: Extrapolation is the use
of a regression line for prediction
outside the range of values of the
explanatory variable used to obtain
the line.
Such predictions are often not
accurate.
Example 5: Does fidgeting keep you slim? Some
people don’t gain weight even when they
overeat. Perhaps fidgeting and other nonexercise activity (NEA) explains why - some
people may spontaneously increase
NEA when fed more. Researchers deliberately
overfed 16 healthy young adults for 8 weeks.
They measured fat gain (in kg) and change in
energy use (in calories) from activity other than
deliberate exercise.
Construct a scatterplot and describe
what you see.
There is a fairly strong negative linear relationship between
the change in NEA and fat gain.
Write the regression equation and interpret
both the slope and the y-intercept.
Fat gain  3.505  .0034( NEA change)
For each increase of 1 calorie in NEA the predicted
fat gain decreases by .0034kg
When there is no change in NEA, the predicted fat
gain is 3.505kg.
Predict the fat gain for an
individual whose NEA increases by
1500 cal.
yˆ  3.505 .0034(1500)
yˆ  1.595
a
b