Quantitative Methods in Business Module Lecture (4): Introduction to Testing of Hypotheses Dr. Mohamed Ramadan moh_ramadan@icloud.com

1. Quantitative Methods
Dr. Mohamed Ramadan
Moh_Ramadan@icloud.com

2. Quantitative Methods
Introduction to
Hypothesis Testing
Lecture (4)

3. Lecture (4) Introduction to Hypothesis Testing
Revision
̅𝑥 ± 𝑍!"#
$
𝜎
𝑛
Confidence Interval at confidence
level 1 − 𝛼
𝒙 = 𝑴𝒐𝒏𝒕𝒉𝒍𝒚 𝑺𝒂𝒍𝒂𝒓𝒚
𝒇𝒙
̅𝑥!
̅𝑥"
̅𝑥#
̅𝑥$
µ

4. Null
Hypothesis
Alternative
Hypothesis
Lecture (4) Introduction to Hypothesis Testing
The Idea ... Concept and Terminologies
𝒙 = 𝑴𝒐𝒏𝒕𝒉𝒍𝒚 𝑺𝒂𝒍𝒂𝒓𝒚
𝒇𝒙
̅𝑥!
̅𝑥"
̅𝑥#
̅𝑥$
The Decision Maker (Prime-
Minister) introduced the following
two hypotheses to be tested:
H0: The Population Mean of
Monthly Salary = 23,100
H1: The Population Mean of
Monthly Salary < 23,100
µ

5. Lecture (4) Introduction to Hypothesis Testing
The Idea ... Concept and Terminologies
𝒙 = 𝑴𝒐𝒏𝒕𝒉𝒍𝒚 𝑺𝒂𝒍𝒂𝒓𝒚
𝒇𝒙
̅𝑥!
̅𝑥"
̅𝑥#
̅𝑥$
The Decision Maker (Prime-
Minister) introduced the following
two hypotheses to be tested:
H0: The Population Mean of
Monthly Salary = 23,100
H1: The Population Mean of
Monthly Salary < 23,100
µ
Decision H0 is True H0 is False
̅𝑥! Reject H0 ✓
̅𝑥# Reject H0 ✓
̅𝑥$ Reject H0 ✕
̅𝑥" Accept H0 ✕

6. α
= 𝑃(𝑡𝑦𝑝𝑒 𝐼 𝑒𝑟𝑟𝑜𝑟) 𝛽
= 𝑃(𝑡𝑦𝑝𝑒 𝐼𝐼 𝑒𝑟𝑟𝑜𝑟)
Lecture (4) Introduction to Hypothesis Testing
Decision H0 is True H0 is False
Reject H0 ✕ ✓
Don’t Reject H0 ✓ ✕
Types of Errors:
Type I error
Type II error
Decision H0 is True
Innocent
H0 is False
Guilty
Reject H0 ✕ ✓
Don’t Reject H0 ✓ ✕
If you are a Judge:
H0: The defendant is innocent
H1: The defendant is guilty
Type II error
Occurs when a null hypothesis
Type I error
Occurs when we a null hypothesis
Types of Errors

7. Lecture (4) Introduction to Hypothesis Testing
The Concept of Testing of Hypotheses
1. There are two hypotheses, the null and the alternative
hypotheses.
2. The procedure begins with the assumption that the null
hypothesis is true.
3. The goal is to determine whether there is enough evidence to
infer that the alternative hypothesis is true.
4. There are two possible decisions:
⦿ Conclude that there is enough evidence to support the
alternative hypothesis. [We reject the null hypothesis]
⦿ Conclude that there is not enough evidence to support the
alternative hypothesis. [We couldn’t reject the null hypothesis]
Hypotheses Testing

8. Lecture (4) Introduction to Hypothesis Testing
The Mechanism of Testing of Hypotheses
𝒙 = 𝑴𝒐𝒏𝒕𝒉𝒍𝒚 𝑺𝒂𝒍𝒂𝒓𝒚
𝒇𝒙
µ
𝐻%: 𝜇 = 23,000
𝐻!: 𝜇 > 23,000
̅𝑥&
α = 𝑃 𝑡𝑦𝑝𝑒 𝐼 𝑒𝑟𝑟𝑜𝑟
α = 𝑃 𝑟𝑒𝑗𝑒𝑐𝑡𝑖𝑛𝑔 𝐻% 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝐻% 𝑖𝑠 𝑡𝑟𝑢𝑒
α = 𝑃 ̅𝑥 > ̅𝑥& 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝐻% 𝑖𝑠 𝑡𝑟𝑢𝑒
α = 𝑃
̅𝑥 − 𝜇
𝜎/ 𝑛
>
̅𝑥& − 𝜇
𝜎/ 𝑛
α = 𝑃 𝑍 > 𝑧'
𝑧' =
̅𝑥& − 𝜇
𝜎/ 𝑛
⋙ ̅𝑥& = 𝑧'
𝜎
𝑛
+ 𝜇
α
Rejection
Region
Acceptance
Region
Hypotheses Testing

9. ̅𝑥& = 𝑧'
𝜎
𝑛
+ 𝜇 = 𝑧%.%)
200
100
+ 23,000
Lecture (4) Introduction to Hypothesis Testing
The Mechanism of Testing of Hypotheses
𝒙 = 𝑴𝒐𝒏𝒕𝒉𝒍𝒚 𝑺𝒂𝒍𝒂𝒓𝒚
𝒇𝒙
µ
𝐻%: 𝜇 = 23,000
𝐻!: 𝜇 > 23,000
̅𝑥&
α = 𝑃 𝑡𝑦𝑝𝑒 𝐼 𝑒𝑟𝑟𝑜𝑟
α = 𝑃 𝑟𝑒𝑗𝑒𝑐𝑡𝑖𝑛𝑔 𝐻% 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝐻% 𝑖𝑠 𝑡𝑟𝑢𝑒
α = 𝑃 ̅𝑥 > ̅𝑥& 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝐻% 𝑖𝑠 𝑡𝑟𝑢𝑒
α = 𝑃
̅𝑥 − 𝜇
𝜎/ 𝑛
>
̅𝑥& − 𝜇
𝜎/ 𝑛
α = 𝑃 𝑍 > 𝑧'
𝑧' =
̅𝑥& − 𝜇
𝜎/ 𝑛
⋙ ̅𝑥& = 𝑧'
𝜎
𝑛
+ 𝜇
̅𝑥 = 23,200 𝑛 = 100 𝜎 = 200 𝛼 = 5%
α = 0.050.5 0.45
Hypotheses Testing

10. Lecture (4) Introduction to Hypothesis Testing
𝒙 = 𝑴𝒐𝒏𝒕𝒉𝒍𝒚 𝑺𝒂𝒍𝒂𝒓𝒚
𝒇𝒙
µ
̅𝑥&
̅𝑥 = 23,200 𝑛 = 100 𝜎 = 200 𝛼 = 5%
̅𝑥& = 𝑧'
𝜎
𝑛
+ 𝜇 = 𝑧%.%)
200
100
+ 23,000
̅𝑥& = 1.65 20 + 23,000 = 23,033
α = 0.050.5 0.45
0.4505
0.05
1.6
Hypotheses Testing

11. Lecture (4) Introduction to Hypothesis Testing
The Mechanism of Testing of Hypotheses
𝒙 = 𝑴𝒐𝒏𝒕𝒉𝒍𝒚 𝑺𝒂𝒍𝒂𝒓𝒚
𝒇𝒙
µ
𝐻%: 𝜇 = 23,000
𝐻!: 𝜇 > 23,000
̅𝑥& = 23,033
α = 𝑃 𝑡𝑦𝑝𝑒 𝐼 𝑒𝑟𝑟𝑜𝑟
α = 𝑃 𝑟𝑒𝑗𝑒𝑐𝑡𝑖𝑛𝑔 𝐻% 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝐻% 𝑖𝑠 𝑡𝑟𝑢𝑒
α = 𝑃 ̅𝑥 > ̅𝑥& 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝐻% 𝑖𝑠 𝑡𝑟𝑢𝑒
α = 𝑃
̅𝑥 − 𝜇
𝜎/ 𝑛
>
̅𝑥& − 𝜇
𝜎/ 𝑛
α = 𝑃 𝑍 > 𝑧'
𝑧' =
̅𝑥& − 𝜇
𝜎/ 𝑛
⋙ ̅𝑥& = 𝑧'
𝜎
𝑛
+ 𝜇
̅𝑥 = 23,200 𝑛 = 100 𝜎 = 200 𝛼 = 5%
̅𝑥& = 𝑧'
𝜎
𝑛
+ 𝜇 = 𝑧%.%)
200
100
+ 23,000
̅𝑥& = 1.65 20 + 23,000 = 23,033
Acceptance
Region
Rejection
Region
Hypotheses Testing
∵ ̅𝑥 > ̅𝑥! ⋙ ∴ 𝑤𝑒 𝑟𝑒𝑗𝑒𝑐𝑡 𝑡ℎ𝑒 𝑛𝑢𝑙𝑙 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 𝑖𝑛 𝑓𝑎𝑣𝑜𝑟 𝑜𝑓 𝐻"

12. Lecture (4) Introduction to Hypothesis Testing
The Mechanism of Testing of Hypotheses (The Simplest Mechanism)
𝐻%: 𝜇 = 23,000
𝐻!: 𝜇 > 23,000
𝑍 =
̅𝑥 − 𝜇
𝜎/ 𝑛
⋙ 𝑤𝑒 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻% 𝑖𝑓 𝑍 > 𝑧'
𝒙 = 𝑴𝒐𝒏𝒕𝒉𝒍𝒚 𝑺𝒂𝒍𝒂𝒓𝒚
𝒇𝒙
𝑧'
α
Rejection
Region
Acceptance
Region
0
Standardized Test Statistic

13. Lecture (4) Introduction to Hypothesis Testing
The Mechanism of Testing of Hypotheses (The Simplest Mechanism)
𝐻%: 𝜇 = 23,000
𝐻!: 𝜇 > 23,000
𝑍 =
̅𝑥 − 𝜇
𝜎/ 𝑛
⋙ 𝑤𝑒 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻% 𝑖𝑓 𝑍 > 𝑧'
𝒙 = 𝑴𝒐𝒏𝒕𝒉𝒍𝒚 𝑺𝒂𝒍𝒂𝒓𝒚
𝒇𝒙
𝑧%.%) = 1.65
Rejection
Region
0
̅𝑥 = 23,200 𝑛 = 100 𝜎 = 200 𝛼 = 5%
𝑧%.%) = 1.65
∵ 𝑍 > 𝑧'
∴ 𝑤𝑒 𝑟𝑒𝑗𝑒𝑐𝑡 𝑡ℎ𝑒 𝑛𝑢𝑙𝑙 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 𝑖𝑛 𝑓𝑎𝑣𝑜𝑟 𝑜𝑓 𝐻!
≡ 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑎𝑙 𝑆𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒
Acceptance
Region
Standardized Test Statistic
𝑍 =
̅𝑥 − 𝜇
𝜎/ 𝑛
=
23,200 − 23,000
200
100
=
200
20
= 10