AP Statistics Section 3.2 ARegression 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 _______________ on the scatterplot.regression line

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 ___________

In this equation, b is the _____, and a is the __________.

bxay ˆ

slopey-intercept

NOTE: You must always define the variables (i.e. and x) in your

regression equation.y

The formulas below allow you to find the value of b depending on the

information given in the problem:

x

y

S

Srb

2xx

yyxxb

i

ii

Once you have computed b, you can then find the value of a using

this equation.

)(xbya

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.

returning) birds of .304(%-31.934 birds new #

Interpreting b: The slope b is the predicted _____________ in the

response variable y as the explanatory variable x increases by

1.

rate of change

Example 2: Interpret the slope of the regression equation for the data on sparrowhawk colonies.

.304by decreases birds new ofnumber

predicted theyear,next colony the the toreturning

birdsadult ofnumber in the 1% of increaseeach For

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 y-intercept of the regression equation for the data on sparrowhawk colonies.

31.934. iscolony in the birds new ofnumber

predicted the0, is returning birds ofpercent When the

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.

486.5ˆ

)87(304.934.31ˆ

y

y

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 non-

exercise 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.

gain.fat andNEA in change the

between iprelationshlinear negative strongfairly a is There

Write the regression equation and interpret both the slope and the y-intercept.

)change (0034.505.3gainFat NEA

.0034kgby decreasesgain fat

predicted NEA thein calorie 1 of increaseeach For

3.505kg. isgain

fat predicted theNEA,in change no is When there

Predict the fat gain for an individual whose NEA increases by

1500 cal.

595.1ˆ

)1500(0034.505.3ˆ

y

y

ab