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Unformatted text preview: u (UMich) Hypothesis Testing 4 / 37 Null and Alternative Hypotheses Examples
1. You are unhappy with your car: you think your new car travels less than 30 mpg. To
get it serviced for free, you want to ﬁnd evidence that MPG < 30. You record your
consumption for a month and build a sample. You challange the claim that
E (MPG ) = µ ≥ 30
H0 :µ ≥ 30
H1 :µ < 30
2. You are an engineer and have a machine in your factory that produces pencils. You
think the machine is broken, and it produces pencils of incorrect lenghts. It supposed
to produce pencils of an average length of 5 inches. You collect a sample and test:
H0 :µ = 5
H1 :µ = 5
KEY: In all hypothesis testing, you look at the evidence and decide whether you have
enough evidence to reject the null hypothesis. If you don’t have it, you fail to reject the
null hypothesis: you don’t accept the null hypothesis. Jury analogy. . .
Utku Suleymanoglu (UMich) Hypothesis Testing 5 / 37 Null and Alternative Hypotheses Type I and II Errors The hypothesis testing we will do is not perfect: there could be mistakes.
Testing Result H0 is True H0 is False Reject H0 Type I Error Correct Fail to Reject H0 Correct Type II Error TYPE I Error: Null Hypothesis is true and you reject it. Probability = α. Signiﬁcance
level. We can choose this.
TYPE II Error: Null Hypothesis is false and you fail to reject it. Probability = β .
Power= 1 − β . This depends on the unknown population parameter. Have limited
control over this. Utku Suleymanoglu (UMich) Hypothesis Testing 6 / 37 Null and Alternative Hypotheses Trial Analogy
Verdict INNOCENT (H0 ) GUILTY Reject (Verdict=Guilty) H0 Type I Error Correct Fail to Reject (Verdict= Not Guilty) H0 Correct Type II Error We set a high standard for convicting people. We assume innocence, then try to ﬁnd
evidence to reject this presumption. We do the same for null hypothesis as well: unless
there is a lot of evidence, we do not reject it.
Type I error: Innocent man gets the chair, Type II error: Murderer walks away. Society
and statisticians try to minimize the probability of Type I error ﬁrst, and demand a lot
evidence to reject an H0 .
Key thing: If we fail to reject H0 , we don’t say “we proved H0 ”, we just don’t have
enough evidence against it. Analogy: if the defendant walks away, his innocence is not
proved, instead his guilt has not been proved with enough evidence.
Utku Suleymanoglu (UMich) Hypothesis Testing 7 / 37 Null and Alternative Hypotheses General Testing Procedure TEST PROCEDURE:
1 Formulate and state null and alternative hypothesis.
2 (Select a signiﬁcan...
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This note was uploaded on 03/17/2014 for the course ECON 404 taught by Professor Staff during the Spring '08 term at University of Michigan.
- Spring '08