Hypothesis_test - Test of Hypotheses Hypothesis: An...

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1 Test of Hypotheses Hypothesis : An assumption, theory, or claim concerning a parameter of a population. For example: Average annual salary of business undergraduate students = $60,000 Average mile per gallon of a U.S.-build SUV = 20 miles / gallon. A name-brand ready-mix pie takes only 5 minutes to make. Hypothesis testing is used to determine whether a statement about the value of a population parameter should or should not be rejected. Null hypothesis , denoted by H 0 , is a tentative assumption about a population parameter. ( 20 ) Alternative hypothesis , denoted by H a , is the opposite of what is stated in the null hypothesis. ( ) Two-tail test : used when the alternative hypothesis is two-sided. 00 0 : : a H H  Reject 0 Reject Do not Reject (Accept) One-tail test: used when the alternative hypothesis is one sided.
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2 1.Lower-tail: 00 0 : : a H H  Reject 0 Do not Reject (Accept) 2. Upper-tail: 0 : : a H H 0 Reject Do not Reject (Accept) Type I Error : denoted by , is rejecting H 0 when it is true. ( False alarm ) Type II Error : denoted by β, is accepting H 0 when it is false. ( Missed detection ) Possible decision Possible states Null hypothesis true Null hypothesis false Accept null hypothesis Correctly accepted Type II error Reject null hypothesis Type I error Correctly rejected
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3 Level of significance : refers to the probability of committing a Type I error; for example : if we use a level of significance of 5%, we risk an error with a probability of 5% or we will be right 95% of the time. Note the relationship with a 95% confidence interval. Types of Hypothesis Tests Mean Proportion Difference between two means Difference between two proportions Hypothesis Testing (3 Methods) σ known σ unknown 1. Standardized Test Statistic / x z n / x t sn 2. Interval Estimation Test z E n ts E n 3. Critical Value Test n z C /  n s t C /  
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4 Tests of Hypotheses : A. Test concerning Means: σ known () or / x x n zz n    B. Tests Concerning Means: σ unknown or / x x n tt s sn df =n-1 C. Test concerning Proportions 00 or (1 ) p x np p p np p D. Differences between Means 12 22 xx z ss nn E. Difference between Proportions 11 ) pp z    
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5 A. Population standard deviation σ known: Lower Tail Test Upper Tail Test Two-tailed Test Hypothesis 00 0 : : a H H  0 : : a H H 0 : : a H H Test Statistic / x z n / x z n / x z n Rejection Rule: p value approach Reject H 0 if p -value ≤ α Reject H 0 if p -value ≤ α Reject H 0 if p -value ≤ α Rejection Rule: Critical Value approach Reject H 0 if zz  Reject H 0 if Reject H 0 if /2 or if Example1 (Two-tail test example) A manufacturer of tires claims (hypothesizes) that they last an average of 25,000 miles each. A random sample of 100 tires was tested and resulted in a x of 24,700 and σ = 4000. Test this hypothesis at a 5% level of significance and assume a two-tail test.
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This note was uploaded on 09/28/2011 for the course STAT METHO 33:623:385 taught by Professor Faridalizadeh during the Spring '11 term at Rutgers.

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Hypothesis_test - Test of Hypotheses Hypothesis: An...

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