Chapter 11Correlation and

Simple Linear Regression

Statistics for Business(Econ)

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Introduction

• In this chapter we employ Regression Analysisto examine the relationship among quantitative variables.

• The technique is used to predict the value of one variable (the dependent variable - y) based on the value of other variables (independent variables x1, x2,…xk.)

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Correlation is a statistical technique that is used to measure and describe a relationship between two variables. The correlation between two variables reflects the degree to which the variables are related.For example:

A researcher interested in the relationship between nutrition and IQ could observe the dietary patterns for a group of children and then measure their IQ scores.A business analyst may wonder if there is any relationship between profit margin and return on capital for a group of public companies.

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A set of n= 6 pairs of scores (X and Y values) is shown in a table and in a scatterplot. The scatterplot allows you to see the relationship between X and Y.

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Positive correlation

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Negative correlation

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Non-linear relationship

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A strong positive relationship, approximately +0.90;

A relatively weak negative correlation, approximately -0.40

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A perfect negative correlation, -1.00

No linear trend, 0.00.

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A demonstration of how one extreme data point (an outrider) can influence the value of a correlation.

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A demonstration of how one extreme data point (an outrider) can influence the value of a correlation.

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Pearson correlation The most common measure of correlation is the Pearson Product Moment Correlation (called Pearson's correlation for short).

=

=

1

1

n

yyxxs

n

iii

xy

yx

xy

ss

sr

correlation coefficient

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The value r2 is called the coefficient of determination because it measures the proportion of variability in one variable that can be determined from the relationship with the other variable. A correlation of r =0.80 (or -0.80), for example, means that r2 =0.64 (or 64%) of the variability in the Y scores can be predicted from the relationship with X.

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Least square fit

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• The linear model

y = dependent variablex = independent variable0 = y-intercept

1 = slope of the line

= error variable

xy 10 xy 10

x

y

0 Run

Rise = Rise/Run

0 and 1 are unknown,therefore, are estimated from the data.

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To calculate the estimates of the coefficientsthat minimize the differences between the data points and the line, use the formulas:

xbyb

s

)Y,Xcov(b

10

2x

1

xbyb

s

)Y,Xcov(b

10

2x

1

The regression equation that estimatesthe equation of the first order linear modelis:

xbby 10ˆ xbby 10ˆ

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• Example 12.1 Relationship between odometer reading and a used car’s selling price.

– A car dealer wants to find the relationship between the odometer reading and the selling price of used cars.

– A random sample of 100 cars is selected, and the data recorded.

– Find the regression line.

Car Odometer Price1 37388 53182 44758 50613 45833 50084 30862 57955 31705 57846 34010 5359

. . .

. . .

. . .

Independent variable x

Dependent variable y

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• Solution– Solving by hand

• To calculate b0 and b1 we need to calculate several statistics first;

;41.411,5y

;45.009,36x

256,356,11n

)yy)(xx()Y,Xcov(

688,528,431n

)xx(s

ii

2i2

x

where n = 100.

533,6)45.009,36)(0312.(41.5411xbyb

0312.688,528,43256,356,1

s

)Y,Xcov(b

10

2x

1

x0312.533,6xbby 10

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4500

5000

5500

6000

19000 29000 39000 49000

OdometerPrice

– Using the computer (see file Xm17-01.xls)

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.806308R Square 0.650132Adjusted R Square0.646562Standard Error151.5688Observations 100

ANOVAdf SS MS F Significance F

Regression 1 4183528 4183528 182.1056 4.4435E-24Residual 98 2251362 22973.09Total 99 6434890

CoefficientsStandard Error t Stat P-valueIntercept 6533.383 84.51232 77.30687 1.22E-89Odometer -0.03116 0.002309 -13.4947 4.44E-24

x0312.533,6y

Tools > Data analysis > Regression > [Shade the y range and the x range] > OK

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This is the slope of the line.For each additional mile on the odometer,the price decreases by an average of $0.0312

4500

5000

5500

6000

19000 29000 39000 49000

Odometer

Price

x0312.533,6y

The intercept is b0 = 6533.

6533

0 No data

Do not interpret the intercept as the “Price of cars that have not been driven”