4. regression analysis1

Regression Analysis
Regression Analysis: the study of the
relationship between variables
Regression Analysis: one of the most
commonly used tools for business analysis
Easy to use and applies to many situations

Regression
• The term regression as a statistical
technique to predict one variable from
another variable.
• It is a measure of the average
relationship between two or more
variables in terms of original units of
data.
• Correlation coefficient is measure of
degree of co-variability between X & Y
but the objective of Regression Analysis
is to study the ‘nature of relationship
between variables’

Types of Regression
• Linear and Non Linear Regression-
• If the given points are plotted on a
graph paper , the points so obtained on
the scatter diagram will more/less
concentrated round a curve called the
curve of Regression.
• If the regression curve is a straight line
then linear otherwise non linear/curved
regression.

Simple & Multiple
Regression
• It is confined with study of two
variables i.e one independent and
other dependent variable.
• It is confined with more than two
variables at a time. I.e two or more
independent variables and one
dependent variable.

Types of variables
• Dependent variable: the single variable
which we wish to estimate/ predict by
the regression model (response variable)
Independent variable: The explanatory
variable(s) used to predict/estimate the
value of dependent variable. (predictor
variable)
• Y = A + B X
• dependent independent

Method of Least Squares
• It states that line should be drawn
through the plotted points in such
manner that the sum of the squares
of the deviations of the actual Y
values from the computed Y values
is the least. In order to obtain a line
which fits the points best
should be minimum.
• LINE OF BEST FIT
2
(Y Yc)

• The straight line is represented by
• Y = a +bX
• In order to determine the values of
a & b, two normal equations are
Y  Na bX
2 XY  aX bX

• Use least square method to estimate, the
increase in sales revenue from an increase of 7.5
percent in advertising expenditure.
Firm Annual % increase in
Advertising
expenditure
Annual % increase
in Sales Revenue
A 1 1
B 3 2
C 4 2
D 6 4
E 8 6
F 9 8
G 11 8
H 14 9

Question For Practice
• The owner of a small garment shop is hopeful that
his sales are rising significantly week by week
.Treating the sales for the previous six weeks as a
typical example of this rising trend, he recorded
them in Rs 1000’s and analyzed the results.
Week 1 2 3 4 5 6
Sales 2.69 2.62 2.80 2.70 2.75 2.81
• Fit a linear regression equation to suggest to him
the weekly srate at which his sales are rising and
Use this equation to estimate expected sales for
the 7th week.

Question for practice
• From the following data obtain two
regression lines:
X Y
6 9
2 11
10 5
4 8
8 7

Regression Lines
• The line which gives the best
estimate of one variable for any
given value of the other variable.
• Y on X-
•


Y  Y  r X 
X
( )
y
x


y
b yx
r
 byx: Regression coefficient
x
of Y on X

X on Y
• The line which gives the best estimate
for the values of X for any specified
value of Y.
• X on Y-


X  X  r Y 
Y


x
b xy
r
y

( )
bxy: Regression coefficient
of X on Y
x
y

• Calculate the regression equations taking
deviations of items from the mean of X
and Y series:
X Y
6 9
2 11
10 5
4 8
8 7

• The following data relate to the scores obtained
by 9 salesman of a company in an intelligence
test and their weekly sales in thousand rupees:
Salesman A B C D E F G H I
Test
Scores
50 60 50 60 80 50 80 40 70
Weekly
sales
30 60 40 50 60 30 70 50 60
(i) Obtain the regression equation of sales on
intelligence test scores of the salesman.
(ii) If the intelligence test score of a salesman is
65 what would be expected weekly sales?

• A survey was conducted to study the
relationship between expenditure(in Rs.) on
accommodation(x) and expenditure on food and
entertainment(y) and the following results were
obtained-
Mean Standard
deviation
Expenditure on
accommodation
65 2.5
Expenditure on
food &
entertainment
67 3.5
Correlation Coefficient =0.8

• Obtain the two regression
equations.
• Estimate the expenditure on food &
entertainment if the expenditure on
accommodation is Rs. 70.
• Estimate expenditure of
accommodation when expenditure on
food & entertainment is Rs. 100.

• In a partially destroyed Lab record of an
analysis of correlation data, the following
results only are legible:
• Variance of X=9.
• Regression Equations
• 8X-10Y+66=0
• 40X-18Y=214
• Find:
• Mean values of X & Y
• Coefficient of correlation between X & Y
• Standard deviation of Y

• Two random variables have the
regression equations:
• 3X+2Y=26
• 6X+Y=31
• Find:
• Mean values of X & Y
• Coefficient of correlation between
X & Y
• Standard deviation of Y if variance
of X is 25.

Case-Let
• The General sales manager of Kiran Enterprises-an
enterprise dealing in the sale of readymade
men’s wear-is toying with the idea of increasing
his sales to Rs. 80,000. On checking the records
of sales during the last 10 years, it was found
that the annual sale proceeds and advertisement
expenditure were highly correlated to the
extent of 0.8.It was further noted that the
annual average sale has been Rs. 45,000 and
annual average advertisement expenditure Rs.
30,000 with a variance of Rs. 1600 and Rs.625 in
advertisement expenditure respectively.How
much expenditure on advertisement would you
suggest the General sales Manager of the
enterprise to incur to meet his target of sales?

4. regression analysis1

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