Terminologies used in the testing of hypothesis are discussed using this ppt. An elaborative definition with suitable examples is given on each slide.

1. Testing of Hypothesis
Mr. Tanuj Kumar Pandey
Assistant Professor (Statistics)
FCBM, AGI, Haldwani
Terminology

2. T e s t i n g o f H y p o t h e s i s
Null Hypothesis: The hypothesis that the observed
difference (between two or more population
characteristics or any specified population
characteristic) is due to sampling or experimental
error (chance) only. So we can say that a Null
Hypothesis is a hypothesis of No Difference. It is
denoted by H0. It has neutral or null attitude
regarding the outcome of the test.
Vaccine – A Vaccine – B
Which vaccine is more effective against COVID-19?
H0: Both vaccines are equally effective.
A test of a statistical hypothesis is a two-action decision problem in which we either reject or fail to
reject the hypothesis based on the analysis of a sample.

3. T e s t i n g o f H y p o t h e s i s
Vaccine – A Vaccine – B
Which vaccine is more effective against COVID-19?
H1: Both vaccines are not equally effective.
Alternative Hypothesis: The rival hypothesis of null hypothesis or hypothesis made according to the claim
or problem or question. It is denoted by H1.
Remark: One cannot say that a null hypothesis is ‘’accepted’’
rather it is correct to say that “cannot be rejected” or “failed to
reject” as it remains to be true based on the statistical evidence
supporting it. Conversely, a null hypothesis that is refuted can be
said to has been “rejected”.
Term origin: Coined by the English Statistician Ronald A.
Fisher better know as Father of Statistics.

4. T e s t i n g o f H y p o t h e s i s

5. T y p e s o f H y p o t h e s i s T e s t
T e s t i n g o f H y p o t h e s i s
Vaccine – A
Which vaccine is more
effective against COVID-19?
Vaccine – B
P o s s i b l e A l t e r n a t i v e H y p o t h e s e s
 H1: Both Vaccines are not equally effective.
 H1: Vaccine A is more effective than Vaccine B.
 H1: Vaccine A is less effective than Vaccine B.
Two Tailed Test: The test which has two possible alternative outcomes after rejecting
the null hypothesis. Symbolically, the test based on
H1: PA ≠ PB (Both vaccines are not equally effective)
One Tailed Test: One possible outcome of the test after rejecting the null hypothesis.
H1 : PA > PB (Vaccine A is more effective than Vaccine B). Right Tailed Test
H1 : PA < PB (Vaccine A is less effective than Vaccine B). Left Tailed Test

6. C r i t i c a l R e g i o n
T e s t i n g o f H y p o t h e s i s
Let us consider a population of 5 coins with weights 1 gm, 2 gm, 3 gm, 4 gm and 5
gm. A random sample of 2 coins has been selected with SRSWOR for testing the
null hypothesis that the mean weight of coins is 3 gm. We have to test
H0 : μ = 3 vs H1 : μ ≠ 3.
Suppose the test criterion is to reject the null hypothesis when sample mean is less
than 2 or more than 4 gm or fail to reject the null hypothesis when sample mean is
between 2 gm to 4 gm.
(1, 2)
(1, 3) (1, 4) (1, 5) (2, 3)
(2, 4) (2, 5) (3, 4), (3, 5)
(4, 5)
S a m p l e S p a c e
Region where
H0 is rejected
even being true.
Region where
H0 is rejected
even being true.
𝑃𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑆𝑎𝑚𝑝𝑙𝑒𝑠 = 𝑁𝐶𝑛
= 5𝐶2
= 10

7. C r i t i c a l R e g i o n
T e s t i n g o f H y p o t h e s i s
The region of rejection of null hypothesis when it is true is that region of the
outcome set where null hypothesis is rejected if the sample point falls in that region
and is called critical region.
10% area to
the right tail
10% area to
the left tail
H1 : μ ≠ 3. Two Tailed Test
H0 : μ = 3.

8. C r i t i c a l R e g i o n
T e s t i n g o f H y p o t h e s i s
10% area to
the left tail
H1 : μ < 3.
Left Tailed Test
The region of rejection of null hypothesis when it is true is that region of the
outcome set where null hypothesis is rejected if the sample point falls in that region
and is called critical region.
The test criterion is to reject the null hypothesis when sample mean is less than 2 gm.
H0 : μ = 3.

9. C r i t i c a l R e g i o n
T e s t i n g o f H y p o t h e s i s
10% area to
the right tail
H1 : μ > 3.
Right Tailed Test
The region of rejection of null hypothesis when it is true is that region of the
outcome set where null hypothesis is rejected if the sample point falls in that region
and is called critical region.
The test criterion is to reject the null hypothesis when sample mean is more than 4 gm.
H0 : μ = 3.

10. Lot is GOOD.
Lot of 10,000
LED Bulbs
Sample of
LED bulbs
Lot of 10,000
LED Bulbs
Sample of
LED bulbs
T y p e s o f E r r o r i n H y p o t h e s i s T e s t i n g
H0 : The defective percentage in the lot is within the allowed limit (Lot is good).
Against H1 : The defective percentage is more than allowed limit (Lot is bad).
Lot is rejected due to higher defectives in sample.
Lot is BAD.
Lot is accepted due to less defectives in sample.
Good LED
Defective LED

11. Assumptions
Reject Lot
(Decision of Rejecting H0)
Accept Lot
(Decision of failing to Reject H0)
Lot is Good (H0 is True)
Wrong Decision
(Type I Error)
Right Decision
(Confidence)
Lot is Bad (H0 is False)
Right Decision
(Power)
Wrong Decision
(Type II Error)
Decision 1: Reject H0 when it is True. (Wrong Decision) ---- Type I Error (False Positive)
Decision 2: Fail to Reject H0 when it is True. (Right Decision) ---- Confidence (True Negative)
Decision 3: Reject H0 when it is False. (Right Decision) ---- Power of the Test (True Positive)
Decision 4: Fail to reject H0 when it is False. (Wrong Decision) ---- Type II Error (False Negative)
T y p e s o f E r r o r i n H y p o t h e s i s T e s t i n g

12. Assumptions
Positive
(Decision of Accepting H1)
Negative
(Decision of failing to Reject H0)
Negative (H0 is True)
False Positive
(Type I Error)
True Negative
(Confidence)
Positive (H1 is True)
True Positive
(Power)
False Positive
(Type II Error)
Decision 1: Reject H0 when it is True. (Wrong Decision) ---- Type I Error (False Positive)
Decision 2: Fail to Reject H0 when it is True. (Right Decision) ---- Confidence (True Negative)
Decision 3: Reject H0 when it is False. (Right Decision) ---- Power of the Test (True Positive)
Decision 4: Fail to reject H0 when it is False. (Wrong Decision) ---- Type II Error (False Negative)
T y p e s o f E r r o r i n H y p o t h e s i s T e s t i n g

13. L e v e l o f S i g n i f i c a n c e & P o w e r o f T e s t
Level of Significance: Probability of committing type I error or size of the critical
region. It is denoted by α.
𝜶 = 𝑃 𝐶𝑜𝑚𝑚𝑖𝑡𝑖𝑛𝑔 𝑇𝑦𝑝𝑒 𝐼 𝐸𝑟𝑟𝑜𝑟
𝜶 = 𝑃 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑛𝑔 𝐻0 𝑤ℎ𝑒𝑛 𝐻0 𝑖𝑠 𝑎𝑐𝑡𝑢𝑎𝑙𝑙𝑦 𝑡𝑟𝑢𝑒
𝜶 = 𝑃 𝑥 ∈ 𝑊
𝐻0
𝜶 =
𝑊
𝐿0𝑑𝑥
Power of Test: Probability of Rejecting H0 when H1 is true. The probability of not
making a Type II error. It is denoted by 1 – β.
1 − β = 𝑃 𝑁𝑜𝑡 𝐶𝑜𝑚𝑚𝑖𝑡𝑖𝑛𝑔 𝑇𝑦𝑝𝑒 𝐼 𝐸𝑟𝑟𝑜𝑟
1 − β = 𝑃 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑛𝑔 𝐻0 𝑤ℎ𝑒𝑛 𝐻1 𝑖𝑠 𝑎𝑐𝑡𝑢𝑎𝑙𝑙𝑦 𝑡𝑟𝑢𝑒
1 − β = 𝑃 𝑥 ∈ 𝑊
𝐻1
1 − β =
𝑊
𝐿1𝑑𝑥

14. C o n f i d e n c e L e v e l
Confidence Level: Probability of Not committing Type I Error.
𝟏 − 𝜶 = 𝑃 𝑁𝑜𝑡 𝐶𝑜𝑚𝑚𝑖𝑡𝑖𝑛𝑔 𝑇𝑦𝑝𝑒 𝐼 𝐸𝑟𝑟𝑜𝑟
𝟏 − 𝜶 = 𝑃 𝑁𝑜𝑡 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑛𝑔 𝐻0 𝑤ℎ𝑒𝑛 𝐻0 𝑖𝑠 𝑎𝑐𝑡𝑢𝑎𝑙𝑙𝑦 𝑡𝑟𝑢𝑒
10% area to
the right tail
10% area to
the left tail
H1 : μ ≠ 3.
Two Tailed Test
H0 : μ = 3.
α = 20%. 1 – α = 80%.
In TOH α remains fixed and we try to minimize β or maximize 1 – β.
(1, 2)
(1, 3) (1, 4) (1, 5) (2, 3)
(2, 4) (2, 5) (3, 4), (3, 5)
(4, 5)
S a m p l e S p a c e
Region where H0 is
rejected even being true.

15. P - v a l u e C o n c e p t
Let us consider a population of 6 coins with weights 1 gm, 2 gm, 3 gm, 4 gm 5gm
and 6 gm. A random sample of 2 coins has been selected using SRSWOR for
testing the null hypothesis that the mean weight of coins is 3.5 gm. We have to test
H0 : μ = 3.5 vs H1 : μ < 3.5
Suppose the test criterion is to reject the null hypothesis when sample mean is less
than 3.5 gm or fail to reject the null hypothesis when sample mean is more than or
equal to 3.5 gm.
S a m p l e S p a c e
(1, 2) (1, 3) (1, 4)
(1, 5) (2, 3) (2, 4)
(1, 6) (2, 5) (2, 6) (3, 4)
(3, 5) (3, 6) (4, 5) (4, 6) (5, 6)
The so called Sampling Distribution of Sample means
𝒙 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
P(𝒙) 1
15
1
15
2
15
2
15
3
15
2
15
2
15
1
15
1
15
α = 40% and 1 – α = 60%.
If we observe 2.5 as the sample mean
then what will be the p-value?

16. P - v a l u e C o n c e p t
Definition: Probability of getting an observed result or more extreme
result under the assumption that the null hypothesis is true.
So, p-value is the
CDF of the sampling
distribution of the test
statistic in case of left
tailed test.
P-value ≅ 27%
• Left (Lower) Tail p-value = CDF
• Right (Upper) Tail p-Value = 1 – CDF
• Two Tailed P-Value = 2 × min 𝐶𝐷𝐹, (1 − 𝐶𝐷𝐹)
• For symmetric sampling distributions:
P-value = 2 × 𝐶𝐷𝐹 𝑜𝑟 2 × 1 − 𝐶𝐷𝐹
Since p-value < α, therefore, we reject the null hypothesis.
If p-value ≤ α, then we reject the null hypothesis otherwise fail to reject it.