Statistical techniques in business ecohomics chap013

Chapter ThirteenLinear Regression and Correlation GOALS When you have completed this chapter, you will be able to: Calculate and interpret the coefficient of correlation, the coefficie

Chapter Thirteen

Linear Regression and

Correlation

GOALS

When you have completed this chapter, you

will be able to:

Calculate and interpret the coefficient of correlation, the coefficient

of determination, and the standard error of estimate.

FOUR

Conduct a test of hypothesis to determine if the population

coefficient of correlation is different from zero

Goals

Linear Regression and Correlation

GOALS

When you have completed this chapter, you

will be able to:

FIVE

Calculate the least squares regression line and interpret the slope and intercept values.

SIX

Construct and interpret a confidence interval and prediction

interval for the dependent variable.

SEVEN

Set up and interpret an ANOVA table

Goals

Correlation Analysis

basis for estimation It

is the predictor variable

Correlation Analysis

Correlation Analysis is a group of statistical techniques to

measure the association between two variables

is a chart that portrays

the relationship between

two variables

being predicted or estimated

Advertising Minutes and $ Sales

0 5 10 15 20 25 30

The Coefficient of Correlation, r

Negative values indicate an

inverse relationship and

positive values indicate a

direct relationship

strength of the relationship between two variables

P earson's r

Also called Pearson’s r and

Pearson’s product moment

Perfect Negative Correlation

Perfect Positive Correlation

Zero Correlation

Strong Positive Correlation

Coefficient of Determination

It is the square of the coefficient of correlation

It ranges from 0 to 1.

It does not give any information on the direction

of the relationship between the variables

proportion of the total variation in the dependent variable

(Y) that is explained or accounted for by the variation in the independent variable (X).

Example 1

Dan Ireland, the student body

president at Toledo State

University, is concerned about

the cost to students of

textbooks He believes there is

a relationship between the

number of pages in the text and

the selling price of the book

To provide insight into the

problem he selects a sample of

eight textbooks currently on

sale in the bookstore Draw a

scatter diagram Compute the

correlation coefficient

Example 1 continued

400 500 600 700 800 60

Mean 625 Mean 79.50

Standard Error 49 Standard Error 4.32

Standard Deviation 139 Standard Deviation 12.21

Sample Variance 19,286 Sample Variance 149.14 Kurtosis -0.55 Kurtosis -0.77 Skewness -0.16 Skewness 0.40

(a)

Page

(b) Price

(c) Page - Mean(Page)

(d) Price-Mean(Price)

(c)*(d)

Did a computed r come from a

population of paired observations

with zero correlation?

t test for the

coefficient of

correlation

t = r  n- 2

different from zero.

T-test of significance of r

use follows the

t distribution.

Step 1

H0 : the correlation in the

population is zero

H1 :The correlation in the

population is not zero.

Computed r = 657 Test the hypothesis that there is

no correlation in the population Use a 02

significance level.

Step 2 Significance level is 02.

Example 1 continued

H0 is not rejected We cannot

reject the hypothesis that there is

no correlation in the population

The amount of association could

test statistic

Regression Analysis

The least squares criterion

is used to determine the equation That is the term

(Y – Y’)2 is minimized

variable (X) to estimate the dependent variable (Y).

It is the estimated Y value when X=0

b is the slope of the line, or the average change

in Y’ for each change of one unit in X

The least squares principle is used to obtain a

and b.

Regression Analysis

b = r sy

sx a = Y – bX

The least squares principle is used to obtain a and

b The equations to determine a and b are:

example 1 that can be

used to estimate the

selling price based on

the number of pages

The sign of the b

value and the sign

The estimated selling price of an

800 page book is $89.61, found by

The Standard Error of Estimate

The formula that is used to compute the standard error:

sy x =   (Y-Y')2

n-2

scatter, or dispersion, of the observed values around the line of regression

Find the standard error of estimate for the problem involving the number of pages in a book and the selling price.

Actual price

(Y)

Estimated price (Y')

Deviation (Y-Y')

Deviation Squared (Y-Y') 2

The Y values are statistically independent This means that in the selection of a sample, the Y values chosen for

a particular X value do not depend on the Y values for

any other X values.

For each value of X, there is a group of Y values, and these Y values are normally distributed

Assumptions Underlying Linear

Regression

The means of these normal

distributions of Y values all

lie on the straight line of

regression

The standard deviations

of these normal distributions are the same

X is the mean of the Xs

n is the number of observations

t is the value of t at n-2 degrees of freedom

The confidence interval for the mean value of Y for a

given value of X is given by:

For our earlier price estimate of $89.61, the

confidence interval, assuming a desired 95%

confidence, is calculated as follows.

Page - Mean(Page) (Page - Mean(Page)) 2

X is 625, the mean of the pages

n is 8, the number of observations

t is 2.447 at 8-2 degrees of freedom and 95% confidence

Prediction Interval

n +  (X-X)2

The prediction interval for an individual value

of Y for a given value of X

Summarizing The Results

The estimated selling price for a book with 800 pages is

The standard error of estimate is $9.94

The 95 percent confidence interval for all books with 800 pages is $89.61 + $14.43 This means the limits are

between $75.18 and $104.04

The 95 percent prediction interval for a particular book with 800 pages is $89.61+ $28.29 The means the limits are between $61.32 and $117.90

These results appear in the following Minitab and Excel outputs

Example 1 revisited