Overview of Hypothesis

1. o Testing and confidence intervals are closely related in a mathematical sense
but the two techniques have different purposes.
o The confidence interval provides an assessment of how accurately we know
the parameter (say, μ)
o The test indicates whether the parameter (μ) could have the hypothesized
value (μo).
Introduction testing of hypothesis:

2. TESTING OF HYPOTHESIS
Testing of hypothesis is a procedure which enables us to
decide on the basis of information obtained from the sample
taken from the population whether to reject or don’t reject
any specified statement or hypothesis regarding the value of
the population parameter in a statistical problem.

3. GENERAL PROCEDURE FOR TESTING HYPOTHESIS :
i Formulate the null and alternative hypotheses.
ii. Decide upon a significance level, α.
Choose an appropriate test statistic.
Determine the CR. The location of the CR depends upon the form of HA. Choose
the location of the critical region on the basis of the direction at which the
inequalities sign points.
If >, choose the right tail as the CR.
If <, choose the left tail as the CR.
If ≠, choose a two-tailed CR.
Compute the value of the test statistic from the sample data.
Reject H if the computed value of test statistic falls in the CR, otherwise don’t
reject H.

4. Steps for Testing of hypothesis about
single population mean
• 1):-Construction of hypotheses
• 1st case
Null hypothesis: 𝐻0: 𝑢 = 𝑢0
alternative hypothesis: 𝐻1: 𝑢 ≠ 𝑢0 two-tailed test
2nd case
Null hypothesis: 𝐻0: 𝑢 ≤ 𝑢0
alternative hypothesis: 𝐻1: 𝑢 > 𝑢0 Right -tailed
3rd case
Null hypothesis: 𝐻0: 𝑢 ≥ 𝑢0
alternative hypothesis: 𝐻1: 𝑢 < 𝑢0 Left-tailed

5. Decide upon the level of significance 𝛼 =?
Test-statistic: 𝑡 =
ҧ
𝑥−𝑢0
ൗ
𝑠
𝑛
=?
Determined the critical region:
Conclusion:
When the condition of critical region hold true.
We will reject 𝐻0, otherwise do not reject.
When reject null hypothesis H0 The critical region will be
𝐻1: 𝑢 ≠ 𝑢0 t≤ −𝑡 Τ
𝛼
2 ,(𝑛−1) and t≥ 𝑡 Τ
𝛼
2 ,(𝑛−1)
𝐻1: 𝑢 > 𝑢0 t≥ 𝑡𝛼,(𝑛−1)
𝐻1: 𝑢 < 𝑢0 t≤ −𝑡𝛼,(𝑛−1)

6. Testing of hypothesis From T-
distribution
• When sample size n≤30 → 𝑇 − 𝑡𝑒𝑠𝑡
• Mean of normal population when
𝛿 𝑢𝑛𝑘𝑛𝑜𝑤𝑛

7. Problem 1st
The average amount of pesticide
that is packed in canes is 6 liters. A
random sample of 10 canes showed
mean amount of pesticide 6.1 liter
with standard deviation of 0.25
liters. Is process out of control?

8. Solution:-
1) Construction of hypotheses:
Ho :  = 6 H1:   6
2) Level of significance  = 5%
3) Test Statistic 𝑡 =
ҧ
𝑥−𝑢0
ൗ
𝑠
𝑛
=
6.1−6
ൗ
0.25
10
= 1.26
Critical region:
𝑡 Τ
𝛼
2 ,(𝑛−1) =𝑡 Τ
0.05
2 ,(10−1) = 𝑡0.025 ,(9) = 2.262
Decision rule:-
1.26≥2.262 1.26 ≤ −2.262
When reject null hypothesis The critical region will be
𝐻1: 𝑢 ≠ 𝑢0 t≤ −𝑡 Τ
𝛼
2 ,(𝑛−1) and t≥ 𝑡 Τ
𝛼
2 ,(𝑛−1)

9. Conclusion:- Condition does not hold true.
So we do not reject H0 so process is in control.

10. • A random sample of 30 wheat forms from
Faisalabad district showed the mean wheat
production of 50 Kg per acre. Can we conclude
that the mean production of wheat from
Faisalabad district is greater than 45 Kg per
acre, if sample variance is un-known to be 9
Kg2. Use 5% level of significance?

11. • 1) Construction of hypotheses
• Ho :   45
• H1:  > 45
• 2) Level of significance
•  = 5%
• 3) Test Statistic
• Test Statistic 𝑡 =
ҧ
𝑥−𝑢0
ൗ
𝑠
𝑛
=
50−45
ൗ
3
30
= 9.13
• 4) Decision Rule:- Reject Ho if tcal  t(n-1)
• 5) Result:-As tcal > t So reject Ho and conclude that the
average wheat production in Faisalabad district is more than
45 Kg per acre.

12. • A random sample of 30 wheat forms from
Faisalabad district showed the mean wheat
production of 50 Kg per acre. Can we conclude
that the mean production of wheat from
Faisalabad district is less than 45 Kg per acre,
if sample variance is known to be 9 Kg2. Use
5% level of significance?

13. • 1) Construction of hypotheses
• Ho :   45
• H1:  > 45
• 2) Level of significance
•  = 5%
• 3) Test Statistic
• Test Statistic 𝑡 =
ҧ
𝑥−𝑢0
ൗ
𝑠
𝑛
=
50−45
ൗ
3
30
= 9.13
•
• 4) Decision Rule:- Reject Ho if t≤ −𝑡𝛼,(𝑛−1)
• 5) Result:-As tcal > t So reject Ho and conclude that the
average wheat production in Faisalabad district is more than
45 Kg per acre.
𝐻1: 𝑢 > 𝑢0 t≤ −𝑡𝛼,(𝑛−1)

14. Assignment
The mean lifetime of bulbs produced by a company has in
past been 1120 hours. A sample of 9 electric light bulbs
recently chosen from a supply of newly produced bulbs
showed a mean lifetime of 1170 hours with a standard
deviation of 120 hours.
1.Test that mean lifetime of the bulbs when lifetime of
bulbs more then 1120? Use 5% & 10 level of significance
2. Also test when lifetime of bulbs less then 1120, Use 5%
& 10 level of significance