Slide 1Learning ObjectivesLearning ObjectivesLearning ObjectivesTerminologyIntroductionSelecting the “Case”Selecting the “Case” – Means (quantitative data)Matched Samples v Independent SamplesThe Repeating LogicExample 1Mini-Case: Delivery Service ProblemMini-Case: Delivery Service ProblemMini-Case: Delivery Service Data - PlanMini-Case: Delivery Service - PlanMini-Case: Delivery Service - PlanMini-Case: Delivery Service - CalculateMini-Case: Delivery Service - ReportStatTools ResultsExample 2Mini-Case: Par Golf, Inc. ProblemMini-Case: Par Golf, Inc. Data - PlanMini-Case: Par Golf, Inc. - PlanMini-Case: Par Golf, Inc. - CalculateMini-Case: Par Golf, Inc. - ReportMini-Case: Par Golf, Inc. – Software ResultsExample 3Mini-Case: Salary Discrimination ProblemMini-case: Salary Discrimination ProblemMini-case: Salary Discrimination - PlanMini-case: Salary Discrimination - PlanMini-case: Salary Discrimination - PlanMini-case: Salary Discrimination - DoMini-case: Salary Discrimination - DoMini-Case: Salary Discrimination - ReportSlide 36To DoComparing Two Groups(µ)1Lecture #4“If you don’t know where you’re going, any road will take you there.” ---- C.S. Lewis (1898 - 1963)Learning Objectives1. Be able to identify the various approaches for comparing two populations with respect to the mean.Paired (matched) SamplesIndependent SamplesPopulation variances knownPopulation variances unknownAssumed equalAssumed unequalLearning Objectives32. Know the parameter and the corresponding sample statistic for each of the approaches for comparing two populations with respect to the mean.3. Understand the sampling distribution for each of the sample statistics used to compare two populations with respect to the mean.4. Understand and be able to calculate a pooled variance estimate of σ2.Learning Objectives45. Be able to construct the appropriate (1 – α)% confidence interval estimate of both µD and µ1 - µ2.6. Be able to conduct appropriate hypothesis tests for both µD and µ1 - µ2.Terminology•Matched (paired) Samples–μD, sD, nD, •Independent Samples–μ1 – μ2 and –Pooled Variance Estimate, sp2–Satterthwaite Approximation5DX21XX 6INTRODUCTIONSelecting the “Case”7•All Classical applications require SRS and Normal distribution for the sample statistic.•Type of data?–Quantitative – use means and standard deviations–Nominal – use proportions•Number of populations (groups)?•How was the data collected? For example, Paired or Independent Samples?•What assumptions can be made? For example, equal or unequal population variances?Selecting the “Case” – Means (quantitative data)8Two GroupsIndependent SamplesParameter: μ1 – μ2Statistic: Population Variances UnknownAssumed ≠Use Satterthwaite t ApproximationAssume =Use t(n1+n2–2)Population Variances knownUse ZPaired Samples Use Di=X1i – X2iParameter: μDStatistic: One Group: See lectures 1 - 321XX DX221221nsnsSppXX22212121nsnsSXX22212121nnXXMatched Samples v Independent SamplesNike wants to see if there is a difference in the durability of 2 sole materials.1. One type of material is placed on one shoe, the other type on the shoe of the same pair. Twenty-five people are given a pair of the shoes to wear for 3 months.2. Twenty-five people are given a pair of shoes soled with material 1, and twenty-five people are given a pair of shoes soled with material 2. The fifty people wear the shoes for 3 months. 9The Repeating Logic10“Reasonable” sampling error: (critical #)(st dev) (critical #)(st error)-4 -3 -2 -1 0 1 2 3 4t Parameterα /21 - α-t* = -tα/2α /2t* = tα/2mSample StatisticmZ-z* = -zα/2z* = zα/2orTwo-sided (symmetric) problemEXAMPLE 111Mini-Case: Delivery Service Problem A Chicago based firm has documents that must be distributed to district offices throughout the United States. Because of the critical information contained in the documents, quick deliveries to the district offices are essential. The firm has decided to select one of two express delivery services that in many instances can provide next-day deliveries to the district offices.12Mini-Case: Delivery Service Problem 13In testing the delivery times of the two services, the firm sends two documents to a sample of ten district offices with one document carried by one delivery service and the other document carried by the second delivery service.Is one service able to deliver in less time than the other, on average?Mini-Case: Delivery Service Data - PlanDistrict OfficeOvernight CourierFlight ExpressDifferenceSeattle 32 25 7Los Angeles 30 24 6Boston 19 15 4Cleveland 16 15 1New York 15 13 2Houston 18 15 3Atlanta 14 15 -1St. Louis 10 8 2Milwaukee 7 9 -2Denver 16 11 5147Mini-Case: Delivery Service - Plan 15•Summarize DataOne Variable Summary OC FE DMean 17.700 15.000 2.700Variance 62.011 31.778 8.456Std. Dev. 7.875 5.637 2.908Median 16.000 15.000 2.000Minimum 7.000 8.000 -2.000Maximum 32.000 25.000 7.000Count 10 10 101st Quartile 14.000 11.000 1.0003rd Quartile 19.000 15.000 5.000Interquartile Range 5.000 4.000 4.000Mini-Case: Delivery Service - Plan 16•Check for Normality, outliers-3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5-3.5-2.5-1.5-0.50.51.52.53.5Q-Q Normal Plot of D Z-ValueStandardized Q-ValueMini-Case: Delivery Service - Calculate 95% Confidence Interval to Estimate the Mean Difference in Delivery Times_________________= (0.62, 4.78)17 DDnDnstxD1;2Mini-Case: Delivery Service - Report With 95% confidence we can state that Overnight Courier will take between 0.62 hours and 4.78 hours longer on average to make the deliveries than Flight Express. 18StatTools ResultsConf. Intervals (Paired-Sample) OC - FESample Size10Sample Mean2.7Sample Std Dev2.908Confidence Level95.0%Degrees of Freedom9Lower Limit0.620Upper Limit4.780 Conf. Intervals (One-Sample) DSample Size10Sample Mean2.700Sample Std Dev2.908Confidence Level (Mean)95.0%Degrees of Freedom9Lower Limit0.620Upper Limit4.78019StatTools > Statistical Inference >Hypothesis Test > Mean/Std. Dev.Analysis Type: One-SampleVariable: DStatTools > Statistical Inference >Hypothesis Test > Mean/Std. Dev.Analysis Type: Paired SampleVariables: OC and FEEXAMPLE 220Mini-Case: Par Golf, Inc. ProblemPar, Inc. is a manufacturer of golf equipment and has recently developed a new golf ball that has been designed for “extra distance.” In a test of driving distance,