This document provides information about regression analysis and linear regression. It defines regression analysis as using relationships between quantitative variables to predict a dependent variable from independent variables. Linear regression finds the best fitting straight line relationship between variables. The simple linear regression equation is given as Y = a + bX, where a and b are estimated parameters calculated from sample data. An example is worked through, showing how to calculate the regression equation from data, graph the relationship, and use the equation to estimate values.

2. Regression Analysis Is a statistical method
which makes use of the relationships between
two or more quantitative variables so that on
variable, called the dependent or response
variable can be predicted with the knowledge
of the values of the other variable, called
independent variable. A regression equation is
a mathematical equation that allow us to
predict values of one dependent variable from
known values of one or more independent
variables.

3. The regression equation is usually expressed
mathematically by a straight line equation
called linear regression line or simple linear
regression. This line will give the best fit to the
relationship of the variables X and Y. The lie
which “best fit” is that line such that when the
differences between the actual values of Y and
the predicted values of Y for each X are
squared and summed, it will result a
minimum.

4. The simple linear regression line is given by the
equation Y = a + bX where Y is the predicted
dependent variable, X is the independent variable, a
and b are the estimates of the parameters of regression
which are calculated from the available sample values
as follows:
N ( XY) – ( X) ( Y)
b=
N( X2 ) – ( X)2
and
a = y – bx where y and x the means of
the sample values of X and Y.

5. Example: Consider the following data:
X 2 4 6 8 10 12
Y 6 7 8 9 10 11
a. Find the equation of regression line
b. Sketch the graph on a scatter diagram
c. Find the point estimate of Y when X = 15

6. Solution (a):
X
Y
X2
Y2
XY
2
4
6
7
4
16
36
49
12
28
6
10
8
9
10
36
64
100
64
81
100
48
72
100
12
11
144
121
132
X2 =
364
Y2 =
XY =
392
8
X = 42
X=7
Y = 51
Y=
8.5
451

7. Then:
6(392) – 42(51)
b=
210
=
6(364) – 422
= 0.5.
420
Thus, a = 8.5 – 0.5(7) = 5. Therefore, the equation
of the regression line is Y = 5 + 0.5X.

8. Solution (b):
Y-Values
14
12
12
10
10
8
8
6
Y-Values
6
4
4
2
2
0
2
4
6
8
10
12
Solution (c) : Substitute x = 15 to the equation Y
= 5 + 0.5X. Thus, Y = 5 + 0.5(15) = 12.5

9. Problem: A study was made by a businessman to
determine the relation between advertising cost
and sales. The following data on 12
commodities were recorded:
Advertisin 30
g cost
(thousand)
15
400
Sales
(thousand)
320 350 490 500 500 530
24
37
42
45
48
40
20
25
20
385
450
390
365 470
Find the estimated regression line and estimate
the sales when X = 43.
35