AP Statistics Section 3.2 ARegression Lines

Linear relationships between two quantitative variables are quite common.

Correlation measures the direction and strength of these relationships. 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.

Who:______________________________What:______________________________ ______________________________Why:_______________________________When, where, how and by whom? The data come from a controlled experiment in which subjects were forced to overeat for an 8-week period. Results of the study were published in Science magazine in 1999.

16 healthy young adultsExp.-change in NEA (cal)Resp.-fat gain (kg)

Do changes in NEA explain weight gain

NEA (calories)

Fat

Gain

(kg)

-100 0 100 200 300 400 500 600 700

8

6

4

2

0

NEA (calories)

Fat

Gain

(kg)

-100 0 100 200 300 400 500 600 700

8

6

4

2

0

Numerical summary: The correlation between NEA

change and fat gain is r = _______

7786.

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

The formula at the right will allow you to find the value of b:

x

y

SS

rb

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

this equation.

)(xbya

We can also find these values on our TI-83/84.

earlierr found way wesame

For this example, the LSL is

or

xy 0034.505.3ˆ

.))((0034.505.3)( calNEAchangekgFatGain

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

response variable y as the explanatory variable x changes.

rate of change

The slope b = -.0034 tells

us that fat gain goes down by .0034 kg for each additional

calorie of NEA.

You cannot say how important a relationship is by looking at how

big the regression slope is.

Interpreting a: The y-intercept a = 3.505 kg is the fat gain estimated by the model if

NEA does not change when a person overeats.

Model: Using the equation above, draw the LSL on your scatterplot.

NEA (calories)

Fat

Gain

(kg)

-100 0 100 200 300 400 500 600 700

8

6

4

2

0

5007.1

10034.

10000340034.

TI 83/84 8:LinReg(a+bx)

GRAPH

121 ,, YLL

ENTERYFunctionVARSY

VARS

1:1:1

Prediction: Predict the fat gain for an individual whose NEA increases

by 400 cal by:

(a) using the graph ___________

(b) using the equation _________

NEA (calories)

Fat

Gain

(kg)

-100 0 100 200 300 400 500 600 700

8

6

4

2

0

Prediction: Predict the fat gain for an individual whose NEA increases

by 400 cal by:

(a) using the graph ___________

(b) using the equation _________

2.2

)400(0034.505.3ˆ y

Prediction: Predict the fat gain for an individual whose NEA increases

by 400 cal by:

(a) using the graph ___________

(b) using the equation _________

2.2

145.2

Predict the fat gain for an individual whose NEA increases by

1500 cal.

595.1ˆ)1500(0034.505.3ˆ

yy

So we are predicting that this individual loses fat when he/she

overeats. What went wrong?

1500 is way outside the range of NEA values in our data

Extrapolation is the use of a regression line for prediction

outside the range of values of the explanatory variable x used to

obtain the line. Such predictions are often not accurate.

ab