General concept for hypohtesis testing

1. GENERAL CONCEPT FOR
HYPOHTESIS TESTING
NADEEM UDDIN
ASSOCIATE PROFESSOR
OF STATISTICS

2. General procedure for hypothesis testing
Following six steps involves in hypothesis testing
Step-I: Hypothesis
State the Null and Alternative hypothesis
from given problem.
Step-II: Level of significance
α = 0.01, 0.05 or any given value.
Step-III: Test Statistic
Choose an appropriate Test Statistic.

3. Step-IV: Critical Region
Determine the critical value and critical
region by observing Alternate hypothesis
with a given level of significance and test of
statistic.
Step-V: Computation
Compute the value of test statistic, using
sample data.
Step-VI: Conclusion
Make decision whether to Accept the Null
hypothesis Or to reject the Null hypothesis.

4. EXPLANATION
Step-I:
Hypothesis:
Any statement, assumption or claim about any
population`s characteristic is called an hypothesis.
For example, Pakistan wins the match against India.
Statistical hypothesis:
Any statement, assumption or claim about any statistical
characteristic of the population is called statistical
hypothesis.
For example,
Pakistan wins the match against India by 30 runs.

5. Simple hypothesis:
If statistical hypothesis is expressed by numerical value by using
“ = ” sign is called simple hypothesis.
For example:
µ = 55inches (Average height of a class students)
Composite hypothesis:
If statistical hypothesis is expressed by a class of numerical values
by using “≠, ˂, ˃, ≤, ≥” signs is called composite hypothesis.
For example:
µ ≠ 55inches (Average height of a class students)
µ ˂ 55inches (Average height of a class students)
µ ˃ 55inches (Average height of a class students)
µ ≤ 55inches (Average height of a class students)
µ ≥ 55inches (Average height of a class students)

6. Null hypothesis:
The null hypothesis, denoted by H0, is usually the
hypothesis that sample observations result purely from
chance.
Alternative hypothesis:
The alternative hypothesis, denoted by H1 or Ha, is the
hypothesis that sample observations are influenced by
some non-random cause.

7. Formulation of Null and Alternate hypothesis
The first step in testing of hypothesis is to established null and
alternative hypothesis. These hypothesis are formulated in such a
way that if one is Accepted the other one is rejected.
The null hypothesis must contains sign of equality like “ = ” , “ ≤
” or “ ≥ ”. and the alternative hypothesis contains “ ≠ ”, “ ˃ ” or “
˂ ”. There are three possible alternative hypothesis to the null
hypothesis stated below
Set-I: H0: θ = θ0 or H0: θ ≤ θ0
H1: θ > θ0 H1: θ > θ0
Set-II: H0: θ = θ0 or H0: θ ≥ θ0
H1: θ < θ0 H1: θ < θ0
Set-III: H0: θ = θ0
H1: θ ≠ θ0

8. Step-II: Errors in Hypothesis Testing
1-Reject H0 when H0 is false (Correct decision)
2-Reject H0 when H0 is true (Wrong decision)
3-Accept H0 when H0 is true (Correct decision)
4-Accept H0 when H0 is false (Wrong decision)

9. Decision Errors:
Two types of errors can result from a hypothesis test.
Type I error:
α = P(Reject 𝐻0 when 𝐻0 is true).
Type II error:
β = P(Accept 𝐻0 when 𝐻0 is false).
Level of significance:
The probability of rejecting the null hypothesis when H0
is true is called a level of significance and denoted by α.

10. Difference between Type-I error and Level of significance:
Type-I error (α) is calculated through distribution and Level
significance is a pre-assumed value.
How to find Type-I Error from distribution.
A random sample of 12 taken from a normal population with a
known variance 16, test the null hypothesis µ = 25 against
alternative hypothesis µ ˃ 25.
Reject 𝐻0 if the sample mean 𝑥 ≥ 26
Solution:
Type-I Error = α = P(Reject 𝐻0 when 𝐻0 is true)

11. Type-I Error = α =P( 𝑥 ≥ 26)
Type-I Error = α =P(
𝑥−𝜇
𝜎
𝑛
≥
26−𝜇
𝜎
𝑛
)
Type-I Error = α = P(𝑍 ≥
26−25
4
12
)
Type-I Error = α =P(𝑍 ≥ 2.16)
Type-I Error = α =1 - P 𝑍 ˂ 2.16
Type-I Error = α = 1- 0.9846
Type-I Error = α = 0.0154
Type-I Error = α = 1.54%

12. How to find Level of significance:
Example-1
Professor: What about your thesis progress?
Student: More than 95 percent I have completed it Sir.
There is 5% pre assumed level of significance.
Example-2
Have you made all the arrangements of your wedding ceremony.
Yes, I have completed it about 99 percent.
There is 1% pre assumed level of significance.
Example-3
Have you finished your preparation for the programme?
Oh yes it is almost 90 percent done.
There is 10% pre assumed level of significance.

13. Step-III: Test Statistic
Test–statistic is a formula used to make the decision whether to
accept or not accept the null hypothesis by applying sample
information on it.
Step-IV: Critical Region
One-Tailed and Two-Tailed Tests:
A test of a statistical hypothesis, where the region of rejection is on
only one side of the sampling distribution, is called a one-tailed
test. For example,
Suppose the null hypothesis states that the mean is less than or
equal to 50. The alternative hypothesis would be that the mean is
greater than 50. The region of rejection would consist of a range of
numbers located on the right side of sampling distribution; that is,
a set of numbers greater than 50.

14. A test of a statistical hypothesis, where the region of rejection is on
both sides of the sampling distribution, is called a two-tailed test.
For example,
Suppose the null hypothesis states that the mean is equal to 50.
The alternative hypothesis would be that the mean is not equal to
50. The region of rejection would consist of a range of numbers
located on both sides of sampling distribution; that is, the region of
rejection would consist partly of numbers that were less than 50
and partly of numbers that were greater than 50.