Section 10.3

Regression

Objective

Given two linearly correlated variables (

x

and

y

), find the linear function (equation) that best describes the trend.

1

Equation of a line

Recall that the equation of a line is given by its

slope

and

y

-intercept

y

=

m

x

+

b 2

Regression

For a set of data (with variables

x

and

y

) that is linearly correlated, we want to find the equation of the line that best describes the trend.

This process is called

Regression 3

Definitions

x

:

The

predictor variable

(Also called the

explanatory variable

or

independent variable

)

y

:

The

response variable

(Also called the

dependent variable

)

Regression Equation

The

equation

that describes the algebraically relationship between the two variables

Regression Line

The

graph

of the regression equation (also called the

line of best fit

or

least squares line

)

4

Definitions

Regression Equation

y

=

b

0

+

b

1

x b

0 : y

-intercept

b

1 :

slope

Regression Line 5

Notation for Regression Equation

y

-intercept Slope Equation

Population



0



1

y

= 

0

+ 

1

x

Sample

b

0

b

1

y

=

b

0

+

b

1

x

6

Requirements

1.

The sample of paired (

x, y

) data is a

random sample

of quantitative data.

2.

Visual examination of the scatterplot shows that the points

approximate a straight-line

pattern.

3.

Any

outliers must be removed

if they are known to be errors. Consider the effects of any outliers that are not known errors.

7

Rounding

b

0

and

b

1

 Round to

three significant digits

 If you use the formulas from the book, do not round intermediate values.

8

Example 1

Refer to the sample data given in

Table 10-1

in the Chapter Problem.

Find the equation of the regression line

in which the explanatory variable (

x

-variable) is the cost of a slice of pizza and the response variable (

y-

variable) is the corresponding cost of a subway fare.

(CPI=Consumer Price Index, not used)

9

Example 1 x : y :

0.15 0.35 1.00 1.25 1.75 2.00

0.15 0.35 1.00 1.35 1.50 2.00

1.

Enter data in

StatCrunch

(columns)

10

Example 1 x : y :

0.15 0.35 1.00 1.25 1.75 2.00

0.15 0.35 1.00 1.35 1.50 2.00

2. Stat

– Regression – Simple Linear

11

Example 1 x : y :

0.15 0.35 1.00 1.25 1.75 2.00

0.15 0.35 1.00 1.35 1.50 2.00

2.

Select

var1

and

var2

(i.e.

x

and

y

values) Click

Calculate

12

Example 1 x : y :

0.15 0.35 1.00 1.25 1.75 2.00

0.15 0.35 1.00 1.35 1.50 2.00

b

0 = 0.0345

b

1 = 0.945 Regression Equation

y

= (0.0345) + (0.945)

x

13

Example 1 Regression Equation

y

= (0.0345) + (0.945)

x

14

Using the Regression Equation for Predictions

1.

Predicted value of

y

is

y = b 0 + b 1

x

2.

Use the regression equation for predictions

only

if the graph of the regression line on the scatterplot confirms that the regression line fits the points reasonably well.

3.

Use the regression equation for predictions

only if the linear correlation coefficient r indicates that there is a linear correlation

two variables.

between the

15

Using the Regression Equation for Predictions

4.

Use the regression line for predictions only if the value of

x

does not go much beyond the scope of the available sample data.

Predicting too far beyond the scope of the available sample data is called

extrapolation

, and it could result in bad predictions.

5.

If the regression equation

does not appear to be useful estimate

for making predictions, the , which is its

best _ sample mean ( y ) 16

Using the Regression Equation for Predictions

Source: www.xkcd.com

17

Strategy for Predicting Values of Y

18

Using the Regression Equation for Predictions

If the regression equation is

not

a good model, the best predicted value of

y

is simply

y

(the mean of the

y

values) Remember, this strategy applies to linear patterns of points in a scatterplot.

19

Definition

For a pair of sample

residual

x

and

y

values, the is the difference between the

observed

sample value of

y

and the

y

-value that is

predicted

by using the regression equation. That is,

Residual

= (observed

y

) – (predicted

y

) =

y

–

y

20

Residuals

21

Definition

A straight line satisfies the

least-squares property

if the sum of the squares of the residuals is the smallest sum possible.

The best possible regression line satisfies this properties (hence why it is also called the

least squares line

)

22

Least Squares Property

sum =

(-5)

2

+ 11

2

+ (-13)

2

+ 7

2 =

364 (any other line would yield a sum larger than 364)

23