1) The document discusses hypothesis testing about linear regression models. It defines linear regression and describes testing hypotheses about the population regression coefficient β. 2) It provides the steps to test hypotheses about β, which include formulating the null and alternative hypotheses, choosing a significance level, determining the test statistic and critical region, computing the regression equation from sample data, and deciding whether to reject or accept the null hypothesis. 3) An example is shown testing the hypothesis H0: β ≤ 0.5 vs Ha: β > 0.5 using sample data, computing the test statistic t and concluding the null hypothesis is rejected since t falls in the critical region.

1. Topic : TESTING OF HYPOTHESIS ABOUT LINEAR REGRESSION
Presented by: Atif Muhammad & Tuahira Raheem
16 17
BS Environmental science 2nd semester
Presented To : Ms. Bushra
Date 18 / 04 / 2018

2. STATISTICS:
The practice or science of collecting and analyzing numerical data in large
quantities,
especially for the purpose of inferring proportions in a whole from those in a
representative sample.
STATISTICAL INFERENCE:
The process of drawing inference about a population on the basis of information
contained in a sample taken from the population is called statistical inference.
Statistical inference is divided into two major areas.

3. 1). ESTIMATION:
It is a procedure by which we obtain an estimate of the truth but unknown value,
a population parameter by using the sample observation from the population.
2). TESTING OF HYPOTHESIS:
It is a procedure which enables us on the basis of information obtained by sampling
whether,
to accept or reject any specified statement or hypothesis regarding the value of the
parameter in a statistical problem.
REGRESSION:
The term regression was introduced by the English biometrician Sir Francis Galton (1822-1911)
to describe a phenomenon which he observed in analyzing the heights of children and their parents.
He found that though tall parents have tall and short parents have short children.

4. The average height of children tends to step back or to regress towards
the average height of all men.
This tendency towards the average height of all men was called a
regression Galton.
Basically, it is a method in which we find the value of x when y is given
and find the value of y when x is given.
These values are approximately value.
Regression analysis:
is a statistical technique that attempts to explore and model the relationship
between two or more variables.

5. For example
Production of wheat depends on quality of seeds
Son depends on father
When the dependent variable depends on two or more dependent
variables are called multiple regression
For example production of wheat depends on seeds, rainfall,
fertilizer etc.
Dependence of the variable upon another variables is called regression
analysis
When the dependent variable depends on only one independent variable
is called simple linear regression

6. Example

8. Hypothesis testing about the linear regression model
Testing hypothesis about β the population regression
co- efficient
1. Formulate the null and alternative hypothesis about B
H β = β0 H1 : β ≠ β0
H0 : β ≤ β H1 : β ˃ β0
H0 : β ≥ β0 H1 : β ˂ β0
2. Decide on the significance level
α = 0.01, α = 0.05 α = 0.1

9. 3. The test statistic to use is
t =
𝑏−𝛽ₒ
𝑆 𝑏
where
4. The critical region is
| t | ≥ 𝑡 2
𝛼
(v) when H1 is β ≠βo
t ≥ 𝑡 𝛼 (v) when H1 is β >βo
t ≤ -𝑡 𝛼 (v) when H1 is β <βo
5. Compute the regression equation
ŷ = a + bx
𝑆 𝑋.𝑌 , 𝑆 𝑏 , and t =
𝑏−𝛽ₒ
𝑆 𝑏
from the sample data
6. Decide as reject H0 if t falls in the critical region accept Ho otherwise

10. Example :
In a linear regression problem the following sums were computed
from a random sample of size 10
Ʃx = 320 Ʃy = 250 Ʃ𝑥2 = 12400 Ʃx.y = 9415 Ʃ ŷ = 7230
1. State null and alternative hypothesis as
Ho: β ≤ 0.5 and Ha : β ˃ 0.5
2. The significance level is set at
α = 0.05
3. The test statistic under Ho is
t =
𝑏−𝛽ₒ
𝑆 𝑏
=
𝑏−0.5
𝑆 𝑏

11. 4. The critical region
t ≥ 𝑡 𝛼 (𝑣)
is t ≥ 𝑡0.05, 8 = 1.86
5. Computation now
b =
𝑛 Ʃ𝑥𝑦− Ʃ𝑥Ʃ𝑦
𝑛 Ʃ𝑥2 −(Ʃ𝑥)2
=
10 9415 − 320 (250)
10 12400 −(320)2
=
14150
21600
b = 0.655

12. a = Ȳ - bx̅
𝑠 𝑦𝑥
2 =
Ʃ(𝑦−ŷ)2
𝑛−2
=
Ʃ𝑌2 −𝑎Ʃ𝑌−𝑏Ʃ𝑋𝑌
𝑛−2
=
7230− 4.04 250 −(0.655)(9415)
10−2
=
53.175
8
𝑆 𝑦𝑥
2 = 6.647
so that 𝑆 𝑦𝑥 = 6.647
= 2.578

13. Ʃ(𝑋 − 𝛸)2
= Ʃ𝑥2
-
(Ʃ𝑥)2
𝑛
= 12400 -
(320)2
10
= 2160
And
𝑆 𝑏 =
𝑆 𝑦.𝑥
Ʃ (𝑥− 𝛸)2
=
2.578
2160
=
2.578
46.476
𝑆 𝑏 = 0.055

14. 6. Conclusion:
since the calculated value of t=2.82 falls in the
critical region so we reject Ho .
We may conclude that there is sufficient evidence to indicate
that the population regression co -efficient is greater than 0.5
t =
𝑏−𝛽𝑜
𝑆 𝑏
t =
0.655 – 0.5
0.055
t = 2.82

15. THANK YOU