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(ii) Lower tailed test:
Review Exercises: Testing HypothesisPlease show all work.No credit for a correct final answer without a valid argu-ment. Use the formula, substitution, answer method whenever possible. Show your workgraphically in all relevant questions.1.A local pizza parlor advertises that their average time for delivery of a pizza iswithin 30 minutes of receipt of the order. The delivery time for a random sample of 6422
orders were recorded, with a sample mean of 34 minutes and a standard deviation of 21minutes.(i) Is there sufficient evidence to conclude that the actual delivery time is larger thanwhat is claimed by the pizza parlor? Useα=.05.H0:Ha:
Graph:Dec:Conclusion:((ii) Test the hypothesis thatHa:µ= 30.2. Answer by True of False . (Circle your choice).
αis fixed and the sample size is increased, thenβwillincrease.23
Chapter 4Small-Sample Tests of HypothesisContents:1. Introduction2. Student’stdistribution3. Small-sample inferences about a population mean4.Small-sample inferences about the difference between two means:IndependentSamples5. Small-sample inferences about the difference between two means: Paired Samples6. Inferences about a population variance7. Comparing two population variances1IntroductionWhen the sample size is small we only deal with normal populations.For non-normal (e.g. binomial) populations different techniques are necessary2Student’stDistributionRECALLFor small samples (n <30) from normal populations, we havez=xµσ/nIfσis unknown, we usesinstead; but we no more have aZdistributionAssumptions.24
1. Sampled population is normal2. Small random sample (n <30)3.σis unknownt=xµs/nProperties of thetDistribution:(i) It hasn1 degrees of freedom (df)(ii) Like the normal distribution it has a symmetric mound-shaped probability distri-bution(iii) More variable (flat) than the normal distribution(iv) The distribution depends on the degrees of freedom.Moreover, asnbecomeslarger,tconverges toZ.(v) Critical values (tail probabilities) are obtained from thettableExamples.(i) Findt0.05,5= 2.015(ii) Findt0.005,8= 3.355(iii) Findt0.025,26= 2.0563Small-Sample Inferences About a Population MeanParameter of interest:µSample data:n,x, sOther information:µ0= target value,αPoint estimator:xEstimator mean:µx=µEstimated standard error:σx=s/nConfidence Interval forµ:x±tα2,n1(sn)Test:H0:µ=µ0Ha: 1)µ > µ0; 2)µ < µ0; 3)µ=µ0.Critical value: eithertα,n1ortα2,n125
T.S. :t=xµ0s/nRR:1) RejectH0ift > tα,n12) RejectH0ift <tα,n13) RejectH0ift > tα2,n1ort <tα2,n1Graph:Decision: 1) if observed value is in RR: “RejectH02) if observed value is not in RR: “Do not rejectH0Conclusion: At 100α% significance level there is (in)sufficient statistical evidence to“favorHa” .Assumptions.

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