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![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](https://image.slidesharecdn.com/19bbst27iqm-lecture4-v1-200306135104/85/Quantitative-Methods-in-Business-Lecture-4-7-320.jpg)
Uploaded byMohamed Ramadan
Quantitative Methods in Business - Lecture (4)
The document provides an introduction to hypothesis testing, explaining the concepts and terminology involved. It outlines the process of testing null and alternative hypotheses, the decision-making framework, types of errors (Type I and Type II), and the statistical significance associated with the findings. Various examples, including calculations related to a population's monthly salary and hypothesis formulation, illustrate the practical application of these concepts.
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Quantitative Methods in Business - Lecture (4)
- 1. Quantitative Methods Dr. MohamedRamadan Moh_Ramadan@icloud.com
- 2. Quantitative Methods Introduction to HypothesisTesting Lecture (4)
- 3. Lecture (4) Introductionto 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) Introductionto 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) Introductionto 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 𝐼𝑒𝑟𝑟𝑜𝑟) 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) Introductionto 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) Introductionto 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) Introductionto 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) Introductionto 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) Introductionto 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) Introductionto 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