# ime261c9 - Alternative hypothesis H1(or Ha Chapter 9 The...

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Chapter 15 / Exercise 15.75
Statistics for Management and Economics + XLSTAT Bind-in
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Chapter 9 1 Statistical Hypotheses A statistical hypothesisis a statement about the parameters of one or more populations. Null hypothesis, H0Alternative hypothesis, H1 (or Ha) Key Points: The null hypothesis is the hypothesis that is always tested. The alternative hypothesis is set up as the opposite of the null hypothesis and represents the conclusion supported if the null is rejected. The null hypothesis always refers to a specified value of the population parameter (such as μ), not a sample statistic (such as X). The statement of the null hypothesis always contains an equal sign regarding the specified value of the parameter. The statement of the alternative hypothesis never contains an equal sign regarding the specified value of the parameter. Chapter 9 2 Hypothesis Test A procedure leading to a decision about a particular hypothesis is called a test of a hypothesis.Rejection of the H0alwaysleads to the acceptance of the H1. Types of Tests:Two-sided alternative hypothesis: H0: μ= μ0H1: μμ0One-sided alternative hypothesis: H0: μ= μ0(H0: μμ0) H1: μ< μ0HYPOTHESES ARE ALWAYS STATEMENTS ABOUT THE POPULATION UNDER STUDY! NOT A STATEMENT ABOUT THE SAMPLE!
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Chapter 15 / Exercise 15.75
Statistics for Management and Economics + XLSTAT Bind-in
Keller Expert Verified
Chapter 9 3 Testing Errors Rejecting H0when it is true is defined as a Type I Error.α= significance level= size of test = P(Type I Error) = P(reject H0when H0is true) Note: confidence level= 1- αFailing to reject the H0when it is false is defined as a Type II error. β= P(Type II Error) = P(fail to reject H0when H0is false) Note: The powerof a statistical test is the probability of rejecting the null hypothesis H0when the alternative hypothesis is true. Power = 1 - βDecision H0is True H0is FalseFail to reject H0No error Type II ErrorReject H0Type I Error No errorChapter 9 4 Example (revisited)
Test the manufacturer's claim. H0: H1: Decision Criteria: 1) Suppose that if 60 x68, we will fail to reject H0. 2) If x< 60 or x> 68, we will reject H0in favor of H1.Notes: (a)Values less than 60 and greater than 68 define the critical region(or regionof rejection) for the test. (b)The interval 60 x68 is called the region of nonrejection. (c)60 and 68 are called the critical values. (d)A decision will be made based on the mean of a single sample taken from the steel.
Chapter 9 5 Example (cont.) The Central Limit Theorem tells us that for a reasonable sample of size n: σ=σμ=μnNXXXXX,~. The probability of making a Type I Error is equal to: P(X< 60 or X> 68) = α. If the H0is true, what is α? What about β? We must set H1equal to a specific value, say μ= 67. Then, given that the alternative, H1, is true: β= P(60 x68) Chapter 9 6 Notes on Testing Errors: 1.Type I and Type II Errors are inversely related. (in one results in the of the other for same sample size) 2.in sample size will both αand β3.Given that H0
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