This document presents information about regression analysis. It defines regression as the dependence of one variable on another and lists the objectives as defining regression, describing its types (simple, multiple, linear), assumptions, models (deterministic, probabilistic), and the method of least squares. Examples are provided to illustrate simple regression of computer speed on processor speed. Formulas are given to calculate the regression coefficients and lines for predicting y from x and x from y.

1. Presentation of Statistics
Submitted to
Mam Uzma
Submitted By
Ali Raza Ansari
13054156-070
Department
Information Technology
UNIVERSITY OF GUJRAT (PAKISTAN)

2. Presentation Topic Is “Regression”

3. Objectives About Regression
 Definition
 Types of Regression
 Types of Dependent Variable
 Assumptions
 Deterministic Model
 Probalistic Model
 Method of Least Square
 Scatter Diagram
 Regression Lines
 Purposes of Regression
 Properties of Regression Line
 Examples

4. Definition :
The dependence of one variable upon variables is called Regression.
“Y = a + bx”

5. Types of regression
There are two types of Regression Variables , which are following.
1. Independent Variable i-e X
2. Dependent Variables i-e Y

6. Types of Dependence of variables
1. Simple Regression
2. Multiple Regression
3. Linear Regression

7. Simple regression
Regression is said to be Simple, if one variable depends upon other Variable.
Examples:
1. Heater depends upon Gas .
2. Motor depends upon Electricity.
Multiple Regression
Regression is said to be Multiple, if two or more variables depends upon other
Variable.
Examples:
1. All Electronic things depends upon Electricity.
2. All Vehicles depends upon Petroleum .

8. Linear regression
Definition :
When the dependency of one variable upon other
variable is represented by a straight line, then Regression is
called Linear Regression.
Example:
Current is directly proportional to voltage

9. A simple linear regression model
Y = α+βx+Є
“Y” is dependent variable.
“x” is independent variable.
“a” is intercept on Y-axis.
“Є” is Error Component.
“β” is Regression Coefficient or Slop
it is also called coefficient of regression.

10. Assumptions :
1. Observations are selected (dependent or independent
variable).
2. Regression function is Linear as , “ α + βx + Є “.
3. Expected values of Error term is zero, “ E ( Є ) = 0”.
4. Variance of Error term is Constant , “ V ( Є ) = δ2”.
5. Error terms are independent of each other, “ E (Єi Єj ) = 0 ” I ≠ j.
6. Error term and x are also zero, “ E (Є x) = 0 “ .
7. Error term is normally distributed with mean zero and Variance
sigma is zero Є~N(0 , δ2).
8. Dependent variable is normally distributed with mean, α + βx
and δ2
1. Є~(α + βx , δ2)

11. Deterministic Model
Definition:
A model in which we can determine a unique value of
dependent variable for each value of independent variable
is called deterministic model.
Formula is “Y = a + bx “.
Examples :
1. Celsius Temperature is “ C = 5 / 9(F – 32) ”.
2. Average is “ Sum of total numbers / Total number (Sum
of n / n)”.
 (Not Error Include)

12. Probabilistic Model
(error Include)
Definition:
A model in which we can’t determine
a unique value of dependent variable for
each value of independent variable is
called probabilistic model.
Formula is “Y = α + βx + Є “.

13. Mothod of Least square
Definition :
Method of least square consists of determining the value of
unknown parameters that minimizes the sum of square of residuals
and it is denoted by,
S.S.R = Σ(Yi – Ŷ)2
Ŷ = a + bx
HERE,
b = nΣxy – ΣxΣy / nΣx2 – (Σx)2
and,
a = Σy – bΣx / n
Range :
- ∞ to + ∞

14. Regression Line
There are two regression lines:
Y=a+bx (y on x)
x=c+dy (x on y)

15. Scatter Diagram
Definition :
Scatter diagram is a graphical picture of sample data.
Consider a random sample of “n” pair of observation
as (x1, y1) ; (x2, y2) …….(xn, yn). These points are plotted
on graph paper taking independent variables on x-axis
and dependent variable on y-axis. Graphical picture so,
obtain is called scatter diagram.
USES:
Scatter diagram is used to judge the relation or
regression such as positive or negative ..

16. Purposes of regression
1.Estimation of unknown parameters (a, b).
2.Prediction of dependent variable , “ Y = a +bx”.
3.Testing of Hypnosis about α and β.
4.Confidence interval of, about α and β.
5.Best procedure available in regression analysis.
6.For Future Prediction .

17. Properties of regression line
1. Regression lines always passes through points (X, Y) .
2. Regression Coefficient independent of origin , byx = bvu if u = x ± a
and v = y ± b .
3. Sum of deviation observe Yi and estimated Ŷ is zero Σ(Yi – Ŷ) = 0.
4. Sum of square of deviation of observed Yi and estimated Ŷ is
Minimum Σ(Yi – Ŷ)2 = min.
5. Sum of observed values is equal to sum of estimated values , Σyi =
ΣŶ .
6. Range of Regression coefficient is - ∞ to + ∞ .

18. Example  Q # Computer Speed Depends upon Processer
SR # Processer speed –’X’ Computer Speed –’Y’
1 128 GHz 20 MB/s
2 768 100
3 378 55
4 1024 185
5 512 80
6 1280 198
X Y x2 y2 xy
128 20 16384 400 2560
768 100 589824 10000 76800
378 55 142884 3025 20790
1024 185 1048576 34225 189440
512 80 262144 6400 40960
1280 198 1638400 39204 253440
4090 638 3698212 93254 583990Total

19. Formula: Y on X
Y = a + bx
a = ў - bхˉ
ў = Σy / n => 638 / 6 = 106.3333
xˉ = Σx / n => 4090 / 6 = 681.6666
b = nΣxy – ΣхΣy / nΣх2 – (Σх)2
b = 6 (583990) – (638)(4090) / 6(3698212)-(4090)2
b = 3503940– 2609420 / 22189272 - 16728100
b=0.1638
a= 106.3333-(0.1638)(681.6666)
a= -5.3237
Prediction: The estimated regression co-efficient y on x b=0.1638 ,which
indicates that the value of y is increase 0.1638 units for a unit increase in x.
Ŷ = -5.3237+ (0.1638)X
Ŷ = -5.3237+(0.1638)(2048)
Ŷ = 330.1387

20. Formula: X on Y
x = c + dy
c = ў - bхˉ
ў = Σy / n => 638 / 6 = 106.3333
xˉ = Σx / n => 4090 / 6 = 681.6666
d = nΣxy – ΣхΣy / nΣy2 – (Σy)2
d = 6 (583990) – (638)(4090) / 6(93254)-(638)2
d = 3503940– 2609420 / 559524 - 407044
d= 1.7241
C = 681.6666-(1.7241)(106.3333)
C = 498.3374
Prediction: The estimated regression co-efficient x on y b=1.7241 ,which
indicates that the value of x is increase 1.7241 units for a unit increase in y.
x = 498.3374+ (1.7241)Y
x = 498.3374+(1.7241)(200)
x = 843.1574

21. Be happy
and
Take care