QUANTITATIVE ANALYSIS FOR

BUSINESSLecture 2

July 5th, 2010

Saksarun (Jay) Mativachranon

INTRO Please turn your mobile phones off or

switch it to silent mode and please do not pick up your calls

Slide will be available atwww.slideshow.com (soon)

Email: saarkman@gmail.com

LINEAR REGRESSION

REGRESSION Regression is used for estimating the

unknown effect of changing one variable over anotherThe variable to be estimated is called

“dependent variable”The changing variable is called

“independent variable”

LINEAR REGRESSION ASSUMPTIONS

1. There is NO relationship between X and Y if 1 equals to 0

2. There is ALWAYS a relationship if 1 does NOT equal to 0

3. The independent Variable (X) is not random4. The expected value of error ( e ) is 0

XY 10

LINEAR REGRESSION ANALYSIS Analyzing the correlation and

directionality of the data Estimating the model Evaluating the validity and usefulness of

the model

USAGE OF REGRESSION Causal analysis Forecasting an effect (of independent

variable to that of dependent variable) Forecasting (trend of) future values

SIMPLE LINEAR REGRESSION True value of slope and intercept are not

known, so we estimate them by using sample data

whereY = dependent variableX = independent variable b0 = intercept (value of Y when X

= 0) b1 = slope of the regression line

XbbY 10 ˆ

^

SCATTER DIAGRAM

EXAMPLE

Linear Regression

SITUATION Company A wants to know the

relationship between the Man Hour of their sales force and their sales number

They have collected their sales data and the man hour put in during the collection period

COMPANY A DATASales of Company A ($) Man Hour (Hour)

6 3

8 4

9 6

5 4

4.5 2

9.5 5

COMPANY A’S SALES SCATTER DIAGRAM 12 –

10 –

8 –

6 –

4 –

2 –

0 –

Sale

s

Man Hour

| | | | | | | |

0 1 2 3 4 5 6 7 8

FINDING THE REGRESSION Company A is trying to predict its sales

from the man hour spent

The line in is the one that minimizes the errors

Y = SalesX = Man Hour

Error = (Actual value) – (Predicted value)

YYe ˆ

DATA MANIPULATION For the simple linear regression model,

the values of the intercept and slope can be calculated using the formulas below

XbbY 10 ˆ

values of (mean) average Xn

XX

values of (mean) average Yn

YY

21 )(

))((

XX

YYXXb

XbYb 10

REGRESSION CALCULATION

Y X (X – X)2 (X – X)(Y – Y)

6 3 (3 – 4)2 = 1 (3 – 4)(6 – 7) = 1

8 4 (4 – 4)2 = 0 (4 – 4)(8 – 7) = 0

9 6 (6 – 4)2 = 4 (6 – 4)(9 – 7) = 4

5 4 (4 – 4)2 = 0 (4 – 4)(5 – 7) = 0

4.5 2 (2 – 4)2 = 4 (2 – 4)(4.5 – 7) = 5

9.5 5 (5 – 4)2 = 1 (5 – 4)(9.5 – 7) = 2.5

ΣY = 42Y = 42/6 = 7

ΣX = 24X = 24/6 = 4

Σ(X – X)2 = 10 Σ(X – X)(Y – Y) = 12.5_ _

_

_ _ _

_ _

REGRESSION CALCULATION (CONT.)

46

246

XX

7642

6YY

25110

51221 .

.)(

))((

XX

YYXXb

24251710 ))(.(XbYb

XY 2512 .ˆ Therefore

RESULTS Company A Sales model

Predicting salesEvery 1 Man-hour, Company A sells $3.25

worth of goods

XY 2512 .ˆ

MEASURING REGRESSION MODEL Regression model can be developed for

any variable Y and X But how do we know the reliability of Y

from variation of X ???

COMPANY A’S SALES MODEL 12 –

10 –

8 –

6 –

4 –

2 –

0 –

Sale

s

Man Hour

| | | | | | | |

0 1 2 3 4 5 6 7 8

ErrorError

MEASURING REGRESSION MODEL (CONT.) How do we know the reliability of Y from

variation of X ???Can we find the average of the errors?

MEASUREMENT OF VARIABILITY SST – Total variability about the mean SSE – Variability about the regression

line SSR – Total variability that is explained

by the model

MEASUREMENT OF VARIABILITY

Sum of the squares total2)( YYSST

Sum of the squared error

22 )ˆ( YYeSSE

Sum of squares due to regression 2)ˆ( YYSSR

An important relationshipSSESSRSST

COMPANY A EXAMPLEY X (Y – Y)2 Y (Y – Y)2 (Y – Y)2

6 3 (6 – 7)2 = 1 2 + 1.25(3) = 5.75 0.0625 1.563

8 4 (8 – 7)2 = 1 2 + 1.25(4) = 7.00 1 0

9 6 (9 – 7)2 = 4 2 + 1.25(6) = 9.50 0.25 6.25

5 4 (5 – 7)2 = 4 2 + 1.25(4) = 7.00 4 0

4.5 2 (4.5 – 7)2 = 6.25

2 + 1.25(2) = 4.50 0 6.25

9.5 5 (9.5 – 7)2 = 6.25

2 + 1.25(5) = 8.25 1.5625 1.563

∑(Y – Y)2 = 22.5 ∑(Y – Y)2 = 6.875 ∑(Y – Y)2 =

15.625

Y = 7 SST = 22.5 SSE = 6.875 SSR = 15.625

^

_

_^

_

_ _^ ^

^

COMPANY A’S VARIABILITYSST = 22.5SSE = 6.875SSR = 15.625

COMPANY A’S SALES MODEL 12 –

10 –

8 –

6 –

4 –

2 –

0 –

Sale

s

Man Hour

| | | | | | | |

0 1 2 3 4 5 6 7 8

Y = 2 + 1.25X^

Y – YY – Y

^

YY – Y^

COEFFICIENT OF DETERMINATION The proportion of variability of Y in the

regression model

COEFFICIENT OF DETERMINATION The coefficient of determination is r2

SSTSSE

SSTSSR

r 12

COMPANY A EXAMPLE

Explanation Over 69% of Y can be predicted by

variation of X

For Company A

69440522

625152 ..

.r

CORRELATION COEFFICIENT The strength of linear relationship

Relationship of Y and X It will always be between +1 and –1 The correlation coefficient is r

CORRELATION COEFFICIENT

**

*

*(a)Perfect Positive

Correlation: r = +1

X

Y

*

* *

*(c) No

Correlation: r = 0

X

Y

* **

** *

* ***

(d)Perfect Negative Correlation: r = –1

X

Y

**

**

* ***

*(b)Positive

Correlation: 0 < r < 1

X

Y

****

*

**

NEXT WEEK Linear Regression

Errors in Regression modelVariance

Mean Square Error Standard Deviation

Testing the Model Multiple Regression