Hypothesis TestingFour parts of statistical testingTest statisticP ValueConclusionOne-Sample Z TestOne-Sample Z Test ExampleSummary of One-Sample Z TestOne-Sample t TestOne-Sample t Test Example 1One-Sample t Test Example 2Summary of One-Sample t TestChapter 8. Testing HypothesesPart 1. Introduction and One-Sample TestsJ.C. WangGoal and ObjectivesIGoal: To learn about Hypotheses Testing Procedures. Tolearn hypothesis-testing methodology as a technique foranalyzing differences and making decisions.IObjectives:IH0vs. H1IType I vs. Type II errorsIOne-tailed vs. Two-tailed testsIp-ValueIUnderstand relationship among α, β, and nIApply methodologyOutlineHypothesis TestingFour parts of statistical testingTest statisticP ValueConclusionOne-Sample Z TestOne-Sample Z Test ExampleSummary of One-Sample Z TestOne-Sample t TestOne-Sample t Test Example 1One-Sample t Test Example 2Summary of One-Sample t TestDefinitionsFour parts of statistical testingIHypothesesITest statisticIp-valueIConclusionStatistical Hypothesesone-tailed testIHypotheses has two parts: Null hypothesis, H0andAlternate hypothesis, H1IKeyword: less than =⇒ left-tailed testH0: µ ≥ µ0vs. H1: µ < µ0, where µ0is a constant.IKeywords: greater than =⇒ right-tailed testH0: µ ≤ µ0vs. H1: µ > µ0, where µ0is a constant.Statistical Hypothesestwo-tailed testIHypotheses has two parts: Null hypothesis, H0andAlternate hypothesis, H1IKeywords: not equal to =⇒ two-tailed testH0: µ = µ0vs. H1: µ 6= µ0, where µ0is a constant.Test StatisticsIThe test statistic is equal to z if the SD is from thepopulation or a process.z =pt.est − H0valueSE, where SE =σ√nIThe test statistic is equal to t if the SD is computed fromthe sample.t =pt.est − H0valueSE, where SE =s√nP Valueone-tailed testp-value comes in two distributions:INormal distributionp-value = normalCDF(|TS|, 9999)IStudent t distributionp-value = tCDF(|TS|, 9999, df )where |TS| is the absolute value of the test statisticP Valuetwo-tailed testp-value comes in two distributions:INormal distributionp-value = 2×normalCDF(|TS|, 9999)IStudent t distributionp-value = 2×tCDF(|TS|, 9999, df )where |TS| is the absolute value of the test statisticDrawing Conclusionin a hypothesis testIf the p-value < α (Significance level), then reject the NullHypothesis (H0): there is evidence that ...; otherwise, do notreject the Null Hypothesis (H0): there is no evidence that ....Note: Significance level = maximum allowable risk ofcommitting type I decision error; p-value = observed risk ofcommitting type I decision erroriClicker Question 8.1 Pre-lectureiClicker Question 8.1 Pre-lectureOne-Sample Z Testcereal box packaging exampleLet’s consider our cereal box example. You are the manager ofthe packaging process at a cereal manufacturing plant. Youwant to determine if the cereal filling process is in control. Theprocess requires no corrective action if the correct amount ofcereal per box is 368 grams. To study this, you decide to take arandom sample of 25 boxes, weigh each one, and thenevaluate the difference between the sample statistic and thehypothesized population parameter by comparing the meanweight from the sample to the expected mean of 368 gramsspecified by the company. The sample mean is 372.5 and theprocess standard deviation is 15. Is there evidence that theweight is different from 368 grams? You have selectedα = 0.05 as your significance level.Cereal Box Packing ExamplecontinuedIGiven: µ0= 368; x = 372.5; σ = 15; n = 25.IWhat are the key words in this problem?Process SD indicates that we will use z for the teststatistic. Is there evidence means that we use ahypothesis test and different from 368 grams meanstwo-tailed alternative.IYou should use this approach to answer the followingquestion.IUse TI calculator z-test because of a process SD.Cereal Box Packing ExamplecontinuedIWhat type of test this is? One-tailed test or two-tailed test?A: Two-tailedIWhat are the hypotheses?A: H0: µ = 368 vs. H1: µ 6= 368IWhat is the significance level α?A: α = 0.05 (and critical value = z.025= 1.96)IWhat is the sample size?A: n = 25IWhat is the standard error (SE)?A : SE =σ√n=15√25= 3Cereal Box Packing ExamplecontinuedIWhat is the test statistic?z =pt.est − H0valueSE=372.5 −3683=4.53= 1.5IWhat is the p-value?A : p-value = 2×normalCDF(1.5, 9999) = 2 × .0668 =.1336IWhat is the conclusion, i.e., is there evidence that theweight is different from 368 grams?A: No, since p-value 6< α = 0.05. Do not reject H0andconclude that there is not enough evidence that the trueaverage weight differs from 368 grams.Cereal Box Packing Exampleusing TI calculatorsIDo this: STAT → TESTS ↓ Z-TEST → STATS ↓ µ:368 ↓σ:15 ↓ x:372.5 ↓ n: 25 ↓ µ:6= µ0↓ CALCULATEIReadout:Z-Testµ 6= 368z=1.5p=.1336x = 372.5n=25One-Sample Z TestsummaryIAssumptionsILarge sample (sample size n ≥ 30, say) or normal dataI‘Known’ population standard deviation σITest statistic (TS): (note: µ0= hypothesized value)z =pt.est − µ0SE=x − µ0σ√nIHypothesis test: one ofILeft-tailed—H0: µ ≥ µ0vs. H1: µ < µ0.IRight-tailed—H0: µ ≤ µ0vs. H1: µ > µ0.ITwo-tailed—H0: µ = µ0vs. H1: µ 6= µ0.One-Sample Z Testsummary, continuedIp-value:#ofTails ×P(Z > |TS|) =#ofTails ×normalCDF(|TS|, 9999)Reject H0if p-value < α.ICritical value:CVal = c × zα/#ofTails= c × invNORM(1 − α/#ofTails),where c = −1 for left-tailed and c = 1, otherwise.IRejection rule using critical value: reject H0I(Left-tailed) if TS < CValI(Right-tailed) if TS > CValI(Two-tailed) if |TS| > CVal. That is, reject H0if TS > CValor TS < −CVal. And we write ±CVal for critical value.One-Sample t TestTV violence exampleNow let’s consider the situation where we must compute thestandard deviation from the data.Suppose you are concerned about the amount of violence onTV. For a Stat 2160 project, you decide to randomly select 10TV programs, watch them, and count the number of “violentscenes” in each. Here is the data you collected:32 12 20 10 4 18 25 26 17 14TV Violence ExamplecontinuedLet’s review the given information before we answer thefollowing question. Is there evidence that the (claim) number of“violent scenes” is at least 21 scenes? What is yourconclusion? How many tails do we have for this test? Assumeα = 0.05.You’re testing against the claim (null hypothesis).TV Violence ExamplecontinuedIHow many tails is this test?A: One; Note the keyword, at least or less than.IWhat are the