Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Chapter 9Tests of Hypotheses for a Single SampleApplied Statistics and Probability for EngineersSixth EditionDouglas C. Montgomery George C. RungerCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Chapter 9 Title and Outline9Tests of Hypotheses for a Single Sample9-1 Hypothesis Testing9-1.1 Statistical Hypotheses9-1.2 Tests of Statistical Hypotheses9-1.3 1-Sided & 2-Sided Hypotheses9-1.4 P-Values in Hypothesis Tests9-1.5 Connection between Hypothesis Tests & Confidence Intervals9-1.6 General Procedure for Hypothesis Tests9-2 Tests on the Mean of a Normal Distribution, Variance Known9-2.1 Hypothesis Tests on the Mean9-2.2 Type II Error & Choice of Sample Size9-2.3 Large-Sample Test9-3 Tests on the Mean of a Normal Distribution, Variance Unknown9-3.1 Hypothesis Tests on the Mean9-3.2 Type II Error & Choice of Sample Size9-4 Tests of the Variance & Standard Deviation of a Normal Distribution.9-4.1 Hypothesis Tests on the Variance9-4.2 Type II Error & Choice of Sample Size9-5 Tests on a Population Proportion9-5.1 Large-Sample Tests on a Proportion9-5.2 Type II Error & Choice of Sample Size 9-6 Summary Table of Inference Procedures for a Single Sample9-7 Testing for Goodness of Fit9-8 Contingency Table Tests9-9 Non-Parametric Procedures9-9.1 The Sign Test9-9.2 The Wilcoxon Signed-Rank Test9-9.3 Comparison to the t-testCHAPTER OUTLINE2Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Learning Objectives for Chapter 9After careful study of this chapter, you should be able to do the following:1. Structure engineering decision-making as hypothesis tests.2. Test hypotheses on the mean of a normal distribution using a Z-test or a t-test.3. Test hypotheses on the variance or standard deviation of a normal distribution.4. Test hypotheses on a population proportion.5. Use the P-value approach for making decisions in hypothesis tests.6. Compute power & Type II error probability. Make sample size selection decisions for tests on means, variances & proportions.7. Explain & use the relationship between confidence intervals & hypothesis tests.8. Use the chi-square goodness-of-fit test to check distributional assumptions.9. Use contingency table tests.Chapter 9 Learning Objectives3Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-1 Hypothesis Testing9-1.1 Statistical HypothesesOne-sided Alternative HypothesesH0: μ = 50 centimeters per second H0: μ = 50 centimeters per secondorH1: μ < 50 centimeters per second H1: μ > 50 centimeters per second4Sec 9-1 Hypothesis TestingA statistical hypothesis is a statement about the parameters of one or more populations.Let H0: μ = 50 centimeters per second and H1: μ ≠ 50 centimeters per secondThe statement H0: μ = 50 is called the null hypothesis.The statement H1: μ ≠ 50 is called the alternative hypothesis.Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-1 Hypothesis TestingTest of a Hypothesis• A procedure leading to a decision about a particular hypothesis• Hypothesis-testing procedures rely on using the information in a. random sample from the population of interest• If this information is consistent with the hypothesis, then we will conclude that the hypothesis is true; if this information is inconsistent with the hypothesis, we will conclude that the hypothesis is false.5Sec 9-1 Hypothesis TestingCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-1.2 Tests of Statistical HypothesesFigure 9-1 Decision criteria for testing H0: = 50 centimeters per second versus H1: 50 centimeters per second.H0: μ = 50 centimeters per secondH1: μ ≠ 50 centimeters per second6Sec 9-1 Hypothesis TestingCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-1 Hypothesis Testing9-1.2 Tests of Statistical HypothesesSometimes the type I error probability is called the significance level, or the -error, or the size of the test.7Sec 9-1 Hypothesis TestingDecisionH0iS TrueH0iS FalseFail to reject H0No errorType II errorReject H0Type I errorNo errorTable 9-1 Decisions in Hypothesis Testing = P(type I error) = P(reject H0when H0is true)β = P(type II error) = P(fail to reject H0when H0is false)Probability of Type I and Type II ErrorCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-1 Hypothesis TestingComputing the Probability of Type I Error8Sec 9-1 Hypothesis TestingThe z-values that correspond to the critical values 48.5 and 51.5 areTherefore P(Z 1.90) P(Z 1.90) 0.028717 0.028717 0.057434which implies 5.74% of all random samples would lead to rejection of the hypothesis H0: μ = 50.)50when5.51()50when5.48( XPXP90.179.0505.51and90.179.0505.4821 zzCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-1 Hypothesis Testing9Sec 9-1 Hypothesis Testing)52when5.515.48( XPThe z-values corresponding to 48.5 and 51.5 when 52 areHence, P(4.43 Z 0.63) P(Z 0.63) P(Z 4.43)= 0.2643 0.0000 0.2643which means that the probability that we will fail to reject the false null hypothesis is 0.2643.63.079.0525.51and43.479.0525.4821 zzComputing the Probability of Type II ErrorCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-1 Hypothesis TestingThe power of a statistical testThe power of a statistical test is the probability of rejecting the null hypothesis H0when the alternative hypothesis is true.• The power is computed as 1 - β, and power can be interpreted as the probability of correctly rejecting a false null hypothesis. • For example, consider the propellant burning rate problem when we are testing H0: μ = 50 centimeters per second against H1: μ not equal 50 centimeters per second . Suppose that the true value of the mean is μ = 52. When n = 10, we found that β = 0.2643, so the power of this test is 1 – β = 1 - 0.2643 = 0.735710Sec 9-1 Hypothesis TestingCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-1 Hypothesis Testing9-1.3 One-Sided and Two-Sided HypothesesTwo-Sided Test:One-Sided Tests:11Sec 9-1 Hypothesis TestingH0: 0H1: ≠ 0H0: 0H1: > 0H0: 0H1: < 0orIn formulating one-sided alternative hypotheses, one should remember that rejecting H0 is always a strong conclusion. Consequently, we
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