Plz, correct the mistakes.

1. Hypothesis Testing
Prepared by:
Towheedul Alam
ID: 1501047

2. Definition
 Any statement or observation or specification about
population parameter or population is called hypothesis.
 Example:
The daily talk time charge of 15 randomly selected customers of Grameenphone
is given below:
110, 118, 130, 140, 142, 146, 112, 100, 95,98,96,122,123,124,130.
Do you think the average talk time of customers in the population is 110?
 To a make logical conclusion, on this situation we use
hypothesis testing.

3. Related Terms
 Parameter: The unknown characteristic of a population.
Such as- µ,σ.
 Statistic: Any function of sample observation. Such as-
𝑥, s.
 Null Hypothesis: is the hypothesis which is picked up
for the test. (denoted by Ho)
 Alternative Hypothesis: is any statement against the
null hypothesis. (denoted by Ha)
If, Ho: µ= µo Then, Ha: (i) µ≠µo or (ii) µ<µO (iii) µ>µO
 Level of significance or Error: The probability level for
which the null hypothesis is rejected even it is true.
(denoted by 𝛼 )

4. Cont.
 Test statistic: The function of sample observations
which is used to verify the null hypothesis.
Three cases for test statistic:
i. If the population variance is known
T statistic: z =
𝒙−µ
σ/√𝒏
ii. If the population variance is unknown but large sample
(≥ 30)
T statistic: z =
𝒙−µ
σ/√𝒏

5. Cont.
iii. If the population variance is unknown and small
sample.(<30)
T statistic: t =
𝒙−µ
𝐬/√𝐧
 Degrees of Freedom (d.f.): The number of independent
values which are used in deriving a test statistic.
Formula: ( n-1) n= Total size of sample.

6. Cont.
 One Tailed and Two Tailed Test:
(i) HA: µ>µO
(ii) HA: µ<µO
(iii) HA: µ≠µO
When, (i) and (ii) condition appears ,it will be called one
tailed test.
When, (iii) condition appears, it will be called two tailed test.
See T- Table

7. T- Table

8. Acceptance Region & Critical
Region
The decision regarding the null hypothesis is made as
follows:
1. If T- statistic ≥ Zά/2,(n-1) , HO is rejected in favour of HA: µ<µO
2. If T- statistic (-)≤ -Zά/2,(n-1) , HO is rejected in favour of HA : µ>µO
3. If T- statistic ≥ Zά/2,(n-1 , HO is rejected in favour of HA : µ≠µO .
Zά/2,(n-1) = Acceptance region. This value can be sought out with the
help of d.f. and level of significance from the T- table.

9. Steps of Hypothesis Testing
 Develop the null hypothesis and alternative hypothesis.
 Specify the level of significance.
 Select the test statistics that will be used to test the
hypothesis.
 Collect the sample data and compute the value of the
test statistic.
 Use the level of significance to determine the critical
value.
 Reject the null hypothesis if the test statistic value is
greater than the critical value.
Go to previous example!

10. Types of Hypothesis Testing
 Test regarding mean
 Test of equality of two means
 Test of equality of two correlated means
 Test regarding proportion
 Test regarding proportion with independent sample
 Test regarding variance etc.

11. Test regarding mean
HO : µ= µo against
HA : µ≠ µo
T- statistic, t =
ẋ−µ
𝐬/√𝐧
z =
ẋ−µ
σ/√𝒏

12. Test of equality of two means
HO : µ1= µ2 against
HA : µ1≠ µ2
T- statistic, Z=
t =
ẋ1− ẋ2
𝑠(
1
𝑛1
+
1
𝑛2
Where, 𝑆2 =
𝑛1 −1 𝑠1
2
𝑛1+𝑛2−2
+
(𝑛2−1)𝑆2
2
𝑛1+𝑛2−2
ẋ1− ẋ2
𝜎1
2
𝑛1
+
𝜎2
2
𝑛2

13. Test Equality of Two Correlated Mean
HO : µ1= µ2 against
HA : µ1≠ µ2
𝑑𝑖= 𝑥𝑖 − 𝑦𝑖 Mean 𝑑= 𝑖=1
𝑛
𝑑 𝑖
𝑛
, Variance, 𝑆 𝑑
2
=
(𝑑 𝑖− 𝑑)2
𝑛−1
Test statistic, t =
𝑑
𝑠 𝑑
√𝑛

14. Test regarding proportion
HO : P= PO (given) against
HA : P≠ PO
Sample proportion, p=
𝑎
𝑛
=
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 𝑖𝑡𝑒𝑚𝑠
𝑠𝑖𝑧𝑒 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒
T- statistic, Z=
𝑝−𝑃 𝑂
𝑃 𝑜𝑄 𝑜
𝑛
where, (Qo =1- Po )
Level of significance 5%

15. Test regarding proportion with independent sample
HO : P1= P2 (given) against
HA : P1 ≠ P2
P1=
𝑎1
𝑛1
= proportion of population-1
P2=
𝑎2
𝑛2
= proportion of population -2
T- statistic, Z =
𝑃1−𝑃2
𝑃𝑄(
1
𝑛1
+
1
𝑛2
)
Where, P =
𝑎1+𝑎2
𝑛1+𝑛2
Q = (1-P)

16. Test regarding variance
HO : 𝜎2
= 𝜎𝑜
2
against
HA : 𝜎2
≠ 𝜎𝑜
2
Since, 𝜇 𝑖𝑠 𝑢𝑛𝑘𝑛𝑜𝑢𝑛 , 𝑠2=
(𝑥 𝑖− 𝑥)2
𝑛−1
T- statistic, χ2 =
(𝑛−1)𝑠2
𝜎 𝑜
2
χ2 is distributed as chi- square with (n-1) d.f.

17. Chi-Square Table