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CHAPTER 10:Hypothesis Testing,One Population Mean or Proportionto accompanyIntroduction to Business Statisticsfourth edition,by Ronald M.WeiersPresentation by Priscilla Chaffe-Stengel Donald N.Stengel 2002 The Wadsworth GroupChapter 10-Learning ObjectivesDescribe the logic of and transform verbal statements into null and alternative hypotheses.Describe what is meant by Type I and Type II errors.Conduct a hypothesis test for a single population mean or proportion.Determine and explain the p-value of a test statistic.Explain the relationship between confidence intervals and hypothesis tests.2002 The Wadsworth GroupNull and Alternative HypothesesNull HypothesesH0:Put here what is typical of the population,a term that characterizes“business as usual”where nothing out of the ordinary occurs.Alternative HypothesesH1:Put here what is the challenge,the view of some characteristic of the population that,if it were true,would trigger some new action,some change in procedures that had previously defined“business as usual.”2002 The Wadsworth GroupBeginning an ExampleWhen a robot welder is in adjustment,its mean time to perform its task is 1.3250 minutes.Past experience has found the standard deviation of the cycle time to be 0.0396 minutes.An incorrect mean operating time can disrupt the efficiency of other activities along the production line.For a recent random sample of 80 jobs,the mean cycle time for the welder was 1.3229 minutes.Does the machine appear to be in need of adjustment?2002 The Wadsworth GroupBuilding HypothesesWhat decision is to be made?The robot welder is in adjustment.The robot welder is not in adjustment.How will we decide?“In adjustment”means =1.3250 minutes.“Not in adjustment”means 1.3250 minutes.Which requires a change from business as usual?What triggers new action?Not in adjustment-H1:1.3250 minutes 2002 The Wadsworth GroupTypes of ErrorNo errorType II error:bType Ierror:aNo errorState of RealityH0 TrueH0 FalseH0 TrueH0 FalseTest Says 2002 The Wadsworth GroupTypes of ErrorType I Error:Saying you reject H0 when it really is true.Rejecting a true H0.Type II Error:Saying you do not reject H0 when it really is false.Failing to reject a false H0.2002 The Wadsworth GroupAcceptable Error for the ExampleDecision makers frequently use a 5%significance level.Use a=0.05.An a-error means that we will decide to adjust the machine when it does not need adjustment.This means,in the case of the robot welder,if the machine is running properly,there is only a 0.05 probability of our making the mistake of concluding that the robot requires adjustment when it really does not.2002 The Wadsworth GroupThe Null HypothesisNondirectional,two-tail test:H0:pop parameter=valueDirectional,right-tail test:H0:pop parameter valueDirectional,left-tail test:H0:pop parameter value Always put hypotheses in terms of population parameters.H0 always gets“=“.2002 The Wadsworth GroupNondirectional,Two-Tail TestsH0:pop parameter=valueH1:pop parameter value 2002 The Wadsworth GroupDirectional,Right-Tail Tests H0:pop parameter valueH1:pop parameter value 2002 The Wadsworth GroupDirectional,Left-Tail TestsH0:pop parameter valueH1:pop parameter value 2002 The Wadsworth GroupThe Logic of Hypothesis TestingStep 1.A claim is made.A new claim is asserted that challenges existing thoughts about a population characteristic.Suggestion:Form the alternative hypothesis first,since it embodies the challenge.2002 The Wadsworth GroupThe Logic of Hypothesis TestingStep 2.How much error are you willing to accept?Select the maximum acceptable error,a a.The decision maker must elect how much error he/she is willing to accept in making an inference about the population.The significance level of the test is the maximum probability that the null hypothesis will be rejected incorrectly,a Type I error.2002 The Wadsworth GroupThe Logic of Hypothesis TestingStep 3.If the null hypothesis were true,what would you expect to see?Assume the null hypothesis is true.This is a very powerful statement.The test is always referenced to the null hypothesis.Form the rejection region,the areas in which the decision maker is willing to reject the presumption of the null hypothesis.2002 The Wadsworth GroupThe Logic of Hypothesis TestingStep 4.What did you actually see?Compute the sample statistic.The sample provides a set of data that serves as a window to the population.The decision maker computes the sample statistic and calculates how far the sample statistic differs from the presumed distribution that is established by the null hypothesis.2002 The Wadsworth GroupThe Logic of Hypothesis TestingStep 5.Make the decision.The decision is a conclusion supported by evidence.The decision maker will:reject the null hypothesis if the sample evidence is so strong,the sample statistic so unlikely,that the decision maker is convinced H1 must be true.fail to reject the null hypothesis if the sample statistic falls in the nonrejection region.In this case,the decision maker is not concluding the null hypothesis is true,only that there is insufficient evidence to dispute it based on this sample.2002 The Wadsworth GroupThe Logic of Hypothesis TestingStep 6.What are the implications of the decision for future actions?State what the decision means in terms of the business situation.The decision maker must draw out the implications of the decision.Is there some action triggered,some change implied?What recommendations might be extended for future attempts to test similar hypotheses?2002 The Wadsworth GroupHypotheses for the ExampleThe hypotheses are:H0:=1.3250 minutes The robot welder is in adjustment.H1:1.3250 minutes The robot welder is not in adjustment.This is a nondirectional,two-tail test.2002 The Wadsworth GroupIdentifying the Appropriate Test StatisticAsk the following questions:lAre the data the result of a measurement(a continuous variable)or a count(a discrete variable)?lIs s known?lWhat shape is the distribution of the population parameter?lWhat is the sample size?2002 The Wadsworth GroupContinuous VariablesContinuous data are the result of a measurement process.Each element of the data set is a measurement representing one sampled individual element.Test of a mean,Example:When a robot welder is in adjustment,its mean time to perform its task is 1.3250 minutes.For a recent sample of 80 jobs,the mean cycle time for the welder was 1.3229 minutes.Note that time to complete each of the 80 jobs was measured.The sample average was computed.2002 The Wadsworth GroupTest of,s Known,Population Normally DistributedTest Statistic:where is the sample statistic.0 is the value identified in the null hypothesis.s is known.n is the sample size.nxzs0m=2002 The Wadsworth GroupTest of,s Known,Population Shape Not Known/Not NormalIf n 30,Test Statistic:If n 30,use a distribution-free test(see Chapter 13).nxzs0m=2002 The Wadsworth GroupTest of,s Unknown,Population Normally DistributedTest Statistic:where is the sample statistic.0 is the value identified in the null hypothesis.s is unknown.n is the sample size degrees of freedom on t are n 1.xxmnst0=2002 The Wadsworth GroupTest of,s Unknown,Population Shape Not Known/Not NormalIf n 30,Test Statistic:If n 30,use a distribution-free test(see Chapter 14).2002 The Wadsworth GroupThe Formal Hypothesis Test for the Example,s KnownI.HypothesesH0:=1.3250 minutesH1:1.3250 minutesII.Rejection Region a=0.05Decision Rule:If z 1.96,reject H0.2002 The Wadsworth GroupThe Formal Hypothesis Test,cont.III.Test StatisticIV.ConclusionSince the test statistic of z=0.47 fell between the critical boundaries of z=1.96,we do not reject H0 with at least95%confidence or at most 5%error.47.000443.00021.0800396.03250.13229.10=m=nxzs 2002 The Wadsworth GroupThe Formal Hypothesis Test,cont.V.ImplicationsThis is not sufficient evidence to conclude that the robot welder is out of adjustment.2002 The Wadsworth GroupDiscrete VariablesDiscrete data are the result of a counting process.The sampled elements are sorted,and the elements with the characteristic of interest are counted.Test of a proportion,pExample:The career services director of Hobart University has said that 70%of the schools seniors enter the job market in a position directly related to their undergraduate field of study.In a sample of 200 of last years graduates,132 or 66%have entered jobs related to their field of study.2002 The Wadsworth GroupTest of p,Sample Sufficiently LargeIf both n p 5 and n(1 p)5,Test Statistic:where p=sample proportion p0 is the value identified in the null hypothesis.n is the sample size.2002 The Wadsworth GroupTest of p,Sample Not Sufficiently LargeIf either n p 5 or n(1 p)5,convert the proportion to the underlying binomial distribution.Note there is no t-test on a population proportion.2002 The Wadsworth 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