Lecture ECO220Y 24 Introduction to Hypothesis Testing
Migiwa Tanaka Reading:11.1
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Outline Outline
Introduction to Hypothesis Testing Testing population mean when the population variance is known
Rejection Region Method p-value Method One-tail vs. two-tail Test
Calculating the probability of a type II error
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Hypothesis Hypothesis Testing
Step1: Identify hypotheses about a population parameter Step2: Collect a sample Step3: Conduct hypothesis testing: Conclude whether the hypotheses set up in step 1 is true or not, using evidence found in the sample.
A. B.
Calculate a test statistic Compare A. with the decision criteria
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Analogy with Criminal Trial (1)
Hypothesis Testing
Criminal Trial
Unknown: Population Parameter Construct hypotheses about population population parameter. Collect a sample from the population Calculate test statistics. Based on the evidence in a sample, test hypothesis.
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Unknown: Defendants Guilt Construct hypotheses about defendants guilt Collect evidence relevant to the crime.
Based on the evidence, test hypothesis
Hypotheses Hypotheses
1.
All hypotheses are constructed in a pair. Null Hypothesis (H0): Initial presumption about the population, which is not based on the data. Status quo. Alternative Hypothesis (H1): The statement that can be proven by the data. New regime. It is also called research hypothesis hypothesis.
2.
Example: H0: Defendant is innocent H1: Defendant is guilty H0: Mean daily demand for gasoline is 1000 gallon 1000 H1: Mean daily demand for gasoline is not 1000 gallon
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Analogy Analogy with Criminal Trial (2)
Juries consider following hypotheses: H0: Defendant is innocent H1: Defendant is guilty Two choices:
1. 2.
Defendant is not guilty Not enough evidence to conclude H0 is true. Do not reject H0 Defendant is guilty Enough evidence beyond reasonable doubt. Reject H0 in favor of H1
Does 1. implies the defendant is proven innocent?
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Construction of Hypotheses: Example 1: Criminal Trial
Current Legal Standard:
Alternative Legal Standard:
Innocent until proven guilty beyond a reasonable doubt
Guilty until proven innocent innocent beyond a reasonable doubt
H0: Defendant is innocent H1: Defendant is guilty What will be the decision if there is no evidence?
H0 : H1 : What will be the decision if there is no evidence?
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What we can do in hypothesis testing:
Reject null hypothesis in favor of alternative(research) hypothesis.
There is enough evidence beyond a reasonable doubt. There is not enough evidence beyond reasonable a doubt to prove the guilty of the defendant.
Do not reject null
What we cannot do:
Accept null hypothesis. (i.e. the defendant is innocent.) (i Prove null hypothesis. (i.e. Prove the innocence of defendant defendant.)
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Type Type I Error and Type II Error
Errors in Decision
H0 is true (innocent) Reject H0 (Convict) Do not reject H0 (Acquit) (Acquit)
H1 is true (guilty)
Type Type I Error
Correct Decision
Correct Decision Type II Error
Type I error is occurs when rejecting true null.
Convict an innocent person. Acquit a guilty person.
Type II error occurs when not rejecting false null.
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Type Type I Error and Type II Error
H0 is true (innocent) Reject H0 (Convict) Do not reject H0 (Acquit) H1 is true (guilty)
Type I Error
Correct Decision
Correct Decision Type II Error
Probability of making type I error= =P(Reject H0 | H0 is true)
This is called level of significance in hypothesis testing.
Probability of making type II error = =P(Do not reject H0 | H1 is true) How and are related?
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Relationship Relationship between and
The probability of type I error( ) and the probability of type II error () are inversely related. Decrease in
Increase in burden of proof Increase the chance of acquittal for both innocent and guilty Increase in (chance of letting guilty person go free)
Decrease in ,
Decrease in burden of proof. Decrease the chance of acquittal for both innocent and guilty defendant. Increase in .
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Construction Construction of Statistical Hypotheses:
Hypothesis: a claim about a population parameter
Null hypothesis
H0: =some value (population mean is 0)
Alternative hypothesis (three ways.)
A. B. C.
H1: some value (population mean is not 0) H1: <some value (population mean is less than 0) H1: >some value (population mean is greater than 0)
A. leads to two-tailed test: When researcher wants to know if the parameter is different from null. B. and C. lead to one-tailed test. : When researcher wants to know if the parameter is greater/less than null.
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Construction of Hypothesis: Example 2: Inventory Management
The retail stores manage their inventory depending on demand for the product. For example, your store stocks canned tomato soup such that it can accommodate 100 cans each week. You want to know if there is an increase in the demand and if you should increase the inventor y level.
H0 : H1 :
You want to know if there is any change in the demand and if you should maintain the current inventory level.
H0 : H1 :
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