Warm Up Determine if the direction and strength of the correlation is as agreed for this class, for the given r value.

1. r=-.89

2. r=.5

3. r=-.2

4. r=0

5. r=-0.72

AP Statistics, Section 3.3, Part 1 2

Section 3.3Linear Regression

AP Statistics

AP Statistics, Section 3.3, Part 1 4

Linear Regression

It would be great to be able to look at multi-variable data and reduce it to a single equation that might help us make predictions

“What would be the predicted number of wins for a team with a 4.0 ERA?”

Wins

Team ERA

30 Major League teams in 2003

AP Statistics, Section 3.3, Part 1 5

Linear Regression

Model equation y is for exact valuey is for predicted value

AP Statistics, Section 3.3, Part 1 6

The Least-Square Regression

Finds the best fit line by trying to minimize the areas formed by the difference of the real data from the predicted data.

AP Statistics, Section 3.3, Part 1 7

The Least-Square Regression (LSRL)

Finds the best fit line by trying to minimize the areas formed by the difference of the real data from the values predicted by the model.

Residual = y-y residuals=0

Largest residual may be an outlier

AP Statistics, Section 3.3, Part 1 8

The Least-Square Regression

Statisticians use a slightly different version of “slope-intercept” form.

There is one point guaranteed to be on the LSRL. (x,y)

y

x

y a bx

sb r

s

a y bx

Know for test equations for test

AP Statistics, Section 3.3, Part 1 9

The Least-Square Regression(using the formula sheet)

Statisticians use a slightly different version of “slope-intercept” form.

There is one point guaranteed to be on the LSRL. (x,y)

1

1

1

ˆ

ˆ o

y y

x x

o

y a bx

y b b x

s sb r or b r

s s

a y bx or b y b x

AP Statistics, Section 3.3, Part 1 11

Fact about least-square regression

Make sure you know which is the explanatory (x) variable and which is the response (y) variable.Switching them gets a different regression

line.

AP Statistics, Section 3.3, Part 1 12

Fact about least-square regression

Regression line always goes through the point (x, y)

The coefficient of correlation (r) explains the strength of the linear relationship

The square of the correlation (r2) is the variation in the values of y that is explained by x.

___%(r2) of the variation of ______ (y) is explained by _____ (x).

Coefficient of explanation

AP Statistics, Section 3.3, Part 1 13

r2 “coefficient of explanation”

In the regression of ERA vs. WINS, we find a r2 value of .4512

We say “45% of the variation in WINS can be explained by ERA”

AP Statistics, Section 3.3, Part 1 14

Outliers vs. Influential Data An outlier is an observation outside the overall

pattern

If an observation is influential it has a large effect on the regression line. Removing the observation markedly changes the calculation.

AP Statistics, Section 3.3, Part 1 15

Outliers vs. Influential Data

Child 19 and 18 are outliers

Child 18 is an influential pointFrom solid to

dashed line

AP Statistics, Section 3.3, Part 1 16

Residuals

It is important to note that the observed value almost never match the predicted values exactly

The difference between the observed value and predicted has a special name: residual

Observed Value (y): 5.3 ERA, 43 Wins

Predicted Value ( ): 5.3 ERA 67.03 Wins

y

Residual:

43-67.03=-24.03ˆy y

AP Statistics, Section 3.3, Part 1 17

Residuals

Observed Value (y): 5.3 ERA, 43 Wins

Predicted Value ( ): 5.3 ERA 67.03 Wins

y

Residual:

43-67.03=-24.03ˆy y

AP Statistics, Section 3.3, Part 1 18

Residual Plots

You can plot the residuals to see if the there is any trends with the quality of the predictive model

AP Statistics, Section 3.3, Part 1 19

Residual Plots

This residual shows no tendencies. It is equally bad throughout.

This suggests that the original relationship is linear.

AP Statistics, Section 3.3, Part 1 20

Not Linear

AP Statistics, Section 3.3, Part 1 21

“Well Distributed”=Linear

AP Statistics, Section 3.3, Part 1 22

Assignment

Section 3.3 Exercises: 3.38(table 1 is on pg.127), 3.40, 3.42, 3.43, 3.46, 3.47, 3.49, 3.53, 3.55, 3.57, 3.61

Chapter 3 Test on

AP Statistics, Section 3.3, Part 1 24

Assignments to end the chapter Chapter Review: 3.63, 3.67, 3.71, 3.73,

3.75, 3.77 Summary Chapter 3 Test

AP Statistics, Section 3.3, Part 1 25

Assignments to end the chapter Chapter 3 Activity (due ?) Chapter Review: 3.63, 3.67, 3.71, 3.73,

3.75, 3.77 Summary Chapter 3 Test