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stat1801_8

stat1801_8 - HypothesisTesting Statistical Hypothesis Null...

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Hypothesis Testing = 118 : 118 : 1 0 μ μ H H Statistical Hypothesis A statement about population Null hypothesis : H 0 Alternative hypothesis : H 1 = 1 : 1 : 2 1 2 0 σ σ H H = 2 1 1 2 1 0 2 : 2 : p p H p p H = 2 2 1 2 2 0 : : y x y x H H σ σ σ σ on. distributi normal a not is Population : on. distributi normal a is Population : 1 0 H H

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Hypothesis Testing Test Procedure (based on sample) leads to rejection or non-rejection of the hypothesis Example : mean IQ of HKU students dropped 118 : vs 118 : Test 1 0 < = μ μ H H " . 113 if Reject " 0 < X H is a reasonable test. Draw a sample 110 = X Reject H 0 Conclude that μ < 118 But we may be wrong !
Two Types of Errors H 0 H 0 H Accept  H 0 Reject  H 0 H true Correct decision Type I error ( H 0  false)  H true Type II error Correct decision ( 29 true | Reject 0 0 H H P = α ( 29 true | Accept 1 0 H H P = β Type I error probability Type II error probability

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Two Types of Errors Example : life time of light bulbs with σ = 300 hours 1240 : vs 1200 : 1 0 = = μ μ H H Test " . 1249 if Reject " 0 X H 100 300 , ~ 2 μ N X Draw a sample with size 100. ( 29 true | Reject 0 0 H H P = α ( 29 1200 | 1249 = = μ α X P - Φ - = 100 300 1200 1249 1 ( 29 0513 . 0 633 . 1 1 = Φ - = ( 29 true | Accept 1 0 H H P = β ( 29 1240 | 1249 = = μ β X P - Φ = 100 300 1240 1249 ( 29 6179 . 0 3 . 0 = Φ =
6179 . 0 = β Two Types of Errors 1200 1240 H 0 H 1 1249 0513 . 0 = α increased β reduced α X of on distributi Sampling

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Type II error Jail an innocence. Choice of  H 0  and  H 1 α β α β α β α β For fix n , we can only control one of the errors. Convention : Control α . Criterion 1 : Type I error as a more serious error. Example : Want to know if a man is guilty. guilty. is He : 0 H guilty. not is He : 1 H Type I error Release a criminal. More serious error guilty. not is He : 0 H guilty. He 1 Type I error Jail an innocence. Type II error Release a criminal. More serious error
Choice of  H 0  and  H 1 Convention : Control α to be small. Criterion 2 : Put what you want to establish in H 1 . P (false rejection of H 0 ) = P (reject H 0 | H 0 true) = α P (false acceptance of H 0 ) = P (accept H 0 | H 1 true) = β May be large Reject H 0 Strong conclusion Accept H 0 Weak conclusion Do not reject

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Choice of  H 0  and  H 1 Criterion 2 : Put what you want to prove in H 1 . Example : Want to show that Brand A is more popular than Brand B. H 0 : B is more popular than A. H 1 : A is more popular than B. If reject H 0 Data shows strong evidence to against H 0 , i.e., high confidence that H 1 is true.
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