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The McGraw-Hill Companies, Inc. 2008McGraw-Hill/Irwin

Chapter 13

Linear Regression and Correlation

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Regression Analysis - Uses

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Some examples.

Is there a relationship between the amount Healthtex spends per

month on advertising and its sales in the month? Can we base an estimate of the cost to heat a home in January on

the number of square feet in the home?

Is there a relationship between the miles per gallon achieved by

large pickup trucks and the size of the engine? Is there a relationship between the number of hours that students

studied for an exam and the score earned?

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Correlation Analysis

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Correlation Analysisis the study of the relationship

between variables. It is also defined as group of techniquesto measure the association between two variables.

A Scatter Diagram is a chart that portrays the relationship

between the two variables. It is the usual first step incorrelations analysis

The Dependent Variable is the variable being predicted orestimated.

The Independent Variable provides the basis for estimation. Itis the predictor variable.

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Regression Example

The sales manager of Copier Sales of

America, which has a large sales forcethroughout the United States and

Canada, wants to determine whetherthere is a relationship between the

number of sales calls made in a month

and the number of copiers sold thatmonth. The manager selects a randomsample of 10 representatives and

determines the number of sales callseach representative made last month

and the number of copiers sold.

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Scatter Diagram

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The Coefficient of Correlation, r

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The Coefficient of Correlation (r) is a measure of the strength of therelationship between two variables. It requires interval or ratio-scaled

data.

It can range from -1.00 to 1.00.

Values of -1.00 or 1.00 indicate perfect and strong correlation. Values close to 0.0 indicate weak correlation.

Negative values indicate an inverse relationship and positive values

indicate a direct relationship.

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Perfect Correlation

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Correlation Coefficient - Interpretation

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Correlation Coefficient - Formula

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Coefficient of Determination

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The coefficient of determination (r2) is the proportion of

the total variation in the dependent variable (Y) that is

explained or accounted for by the variation in the

independent variable (X). It is the square of the coefficient

of correlation. It does not give any information on the direction of the

relationship between the variables.

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Correlation Coefficient - Example

Using the Copier Sales ofAmerica data which a

scatterplot was developed

earlier, compute the

correlation coefficient andcoefficient of determination.

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It is positive relationship

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Coefficient of Determination (r2) - Example

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The coefficient of determination, r2 ,is 0.576,found by (0.759)2

This is a proportion or a percent; we can say that

57.6 percent of the variation in the number ofcopiers sold is explained, or accounted for, by the

variation in the number of sales calls.

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Linear Regression Model

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Computing the Slope of the Line

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Computing the Y-Intercept

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Regression Analysis Least Squares

Principle

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The least squares principle is used to obtain aand b.

The equations to determine a and b are:

bn XY X Y

n X X

a Yn

b Xn

=

=

( ) ( )( )

( ) ( )

2 2

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Illustration of the Least Squares Regression

Principle

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Regression Equation - Example

Recall the example involving Copier Sales

of America. The sales manager gatheredinformation on the number of sales calls

made and the number of copiers soldfor a random sample of 10 sales

representatives. Use the least squaresmethod to determine a linear equationto express the relationship between the

two variables.

What is the expected number of copierssold by a representative who made 20

calls?

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Finding the Regression Equation - Example

6316.42

)20(1842.19476.18

1842.19476.18

:isequationregressionThe

^

^

^

^

=

+=

+=

+=

Y

Y

XY

bXaY

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Plotting the Estimated and the Actual Ys

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End of Chapter 13

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chap13-19-12-11

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