Business Statistics A First Course (3rd Edition)

1、Business Statistics: A First Course (3rd Edition)Chapter 8Hypothesis Tests for Numerical Data from Two or More Samples1Chapter TopicsComparing Two Independent SamplesIndependent samples Z Test for the difference in two meansPooled-variance t Test for the difference in two meansF Test for the Differe

2、nce in Two VariancesComparing Two Related SamplesPaired-sample Z test for the mean differencePaired-sample t test for the mean difference2The Completely Randomized Design: One-Way Analysis of VarianceANOVA AssumptionsF Test for Difference in More than Two MeansThe Tukey-Kramer ProcedureChapter Topic

3、s(continued)3Comparing Two Independent SamplesDifferent Data SourcesUnrelatedIndependentSample selected from one population has no effect or bearing on the sample selected from the other populationUse the Difference between 2 Sample MeansUse Z Test or Pooled-Variance t Test4Independent Sample Z Test

4、 (Variances Known)AssumptionsSamples are randomly and independently drawn from normal distributionsPopulation variances are knownTest Statistic 5Independent Sample (Two Sample) Z Test in EXCELIndependent Sample Z Test with Variances KnownTools | Data Analysis | z-test: Two Sample for Means6Pooled-Va

5、riance t Test (Variances Unknown)AssumptionsBoth populations are normally distributedSamples are randomly and independently drawnPopulation variances are unknown but assumed equalIf both populations are not normal, need large sample sizes7Developing the Pooled-Variance t TestSetting Up the Hypothese

6、sH0: m 1 m 2 H1: m 1 m 2 H0: m 1 -m 2 = 0 H1: m 1 - m 2 0H0: m 1 = m 2 H1: m 1 m 2 H0: m 1 m 2 H0: m 1 - m 2 0 H1: m 1 - m 2 0H0: m 1 - m 2 0 H1: m 1 - m 2 0ORORORLeft TailRight TailTwo Tail H1: m 1 m 28Developing the Pooled-Variance t TestCalculate the Pooled Sample Variance as an Estimate of the C

7、ommon Population Variance(continued)9Developing the Pooled-Variance t TestCompute the Sample Statistic(continued)Hypothesized difference10Pooled-Variance t Test: ExampleYoure a financial analyst for Charles Schwab. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? Y

8、ou collect the following data: NYSE NASDAQNumber 21 25Sample Mean 3.27 2.53Sample Std Dev 1.30 1.16Assuming equal variances, isthere a difference in average yield (a = 0.05)? 1984-1994 T/Maker Co.11Calculating the Test Statistic12SolutionH0: m1 - m2 = 0 i.e. (m1 = m2)H1: m1 - m2 0 i.e. (m1 m2)a = 0.

9、05df = 21 + 25 - 2 = 44Critical Value(s):Test Statistic: Decision:Conclusion:Reject at a = 0.05There is evidence of a difference in means.t02.0154-2.0154.025Reject H0Reject H0.0252.0313p -Value Solutionp-Value 2(p-Value is between .02 and .05) (a = 0.05). Reject.02.03ZRejecta 22.0154is between .01 a

10、nd .025Test Statistic 2.03 is in the Reject RegionReject-2.0154=.02514Pooled-Variance t Test in PHStat and ExcelIf the Raw Data are AvailableTools | Data Analysis | t-Test: Two Sample Assuming Equal VariancesIf only Summary Statistics are AvailablePHStat | Two-Sample Tests | t Test for Differences i

11、n Two Means.15Solution in EXCELExcel Workbook that Performs the Pooled-Variance t Test 16Confidence Interval Estimate for of Two Independent GroupsAssumptionsBoth populations are normally distributedSamples are randomly and independently drawnPopulation variances are unknown but assumed equalIf both

12、 populations are not normal, need large sample sizes Confidence Interval Estimate:17Example 1984-1994 T/Maker Co.Youre a financial analyst for Charles Schwab. You collect the following data: NYSE NASDAQNumber 21 25Sample Mean 3.27 2.53Sample Std Dev 1.30 1.16You want to construct a 95% confidence in

13、terval for the difference in population average yields of the stocks listed on NYSE and NASDAQ.18Example: Solution19Solution in ExcelAn Excel Spreadsheet with the Solution: 20F Test for Difference in Two Population VariancesTest for the Difference in 2 Independent PopulationsParametric Test Procedur

14、eAssumptionsBoth populations are normally distributedTest is not robust to this violationSamples are randomly and independently drawn 21The F Test Statistic = Variance of Sample 1 n1 - 1 = degrees of freedom n2 - 1 = degrees of freedomF 0 = Variance of Sample 222HypothesesH0: s12 = s22 H1: s12 s22 T

15、est StatisticF = S12 /S22Two Sets of Degrees of Freedomdf1 = n1 - 1; df2 = n2 - 1Critical Values: FL( ) and FU( ) FL = 1/FU* (*degrees of freedom switched)Developing the F TestReject H0Reject H0a/2a/2Do NotRejectF 0FLFU n1 -1, n2 -1 n1 -1 , n2 -123F Test: An ExampleAssume you are a financial analyst

16、 for Charles Schwab. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data: NYSE NASDAQNumber 2125Mean3.272.53Std Dev1.301.16Is there a difference in the variances between the NYSE & NASDAQ at the a = 0.05 level? 1984-1994 T/Maker Co.24F Test:

17、 Example SolutionFinding the Critical Values for a = .05 25F Test: Example Solution H0: s12 = s22 H1: s12 s22 a = .05 df1 = 20 df2 = 24 Critical Value(s):Test Statistic: Decision:Conclusion:Do not reject at a = 0.05There is insufficient evidence to prove a difference in variances.0F2.330.415.025Reje

18、ctReject.0251.2526F Test in PHStatPHStat | Two-Sample Tests | F Test for Differences in Two VariancesExample in Excel Spreadsheet27F Test: One-TailH0: s12 s22H1: s12 s22Reject a = .05F0F0Reject a = .05 a = .05orDegrees of freedom switched28Comparing Two Related SamplesTest the Means of Two Related S

19、amplesPaired or matchedRepeated measures (before and after)Use difference between pairsEliminates Variation between Subjects 29Z Test for Mean Difference (Variance Known)AssumptionsBoth populations are normally distributedObservations are paired or matchedVariance KnownTest Statistic 30t Test for Me

20、an Difference (Variance Unknown)AssumptionsBoth populations are normally distributedObservations are matched or pairedVariance unknownIf population not normal, need large samplesTest Statistic31Existing System (1) New Software (2) Difference Di9.98 Seconds 9.88 Seconds .109.88 9.86 .029.84 9.75 .099

21、.99 9.80 .199.94 9.87 .079.84 9.84 .009.86 9.87 - .0110.12 9.98 .149.90 9.83 .079.91 9.86 .05Paired-Sample t Test: ExampleAssume you work in the finance department. Is the new financial package faster (a=0.05 level)? You collect the following processing times:32Paired-Sample t Test: Example Solution

22、Is the new financial package faster (0.05 level)? .072D =H0: mD 0 H1: mD 0 a =.05Test StatisticCritical Value=1.8331 df = n - 1 = 9Reject a =.051.8331Decision: Reject H0t Stat. in the rejection zone.Conclusion: The new software package is faster.3.6633Paired-Sample t Test in EXCELTools | Data Analys

23、is | t-test: Paired Two Sample for MeansExample in Excel Spreadsheet34General Experimental SettingInvestigator Controls One or More Independent VariablesCalled treatment variables or factorsEach treatment factor contains two or more groups (or levels)Observe Effects on Dependent VariableResponse to

24、groups (or levels) of independent variableExperimental Design: The Plan Used to Test Hypothesis35Completely Randomized DesignExperimental Units (Subjects) are Assigned Randomly to GroupsSubjects are assumed homogeneousOnly One Factor or Independent VariableWith 2 or more groups (or levels)Analyzed b

25、y One-way Analysis of Variance (ANOVA)36Factor (Training Method)Factor Levels(Groups)Randomly Assigned UnitsDependent Variable(Response)21 hrs17 hrs31 hrs27 hrs25 hrs28 hrs29 hrs20 hrs22 hrsRandomized Design Example37One-way Analysis of VarianceF TestEvaluate the Difference among the Mean Responses

26、of 2 or More (c ) PopulationsE.g. Several types of tires, oven temperature settingsAssumptionsSamples are randomly and independently drawnThis condition must be metPopulations are normally distributedF Test is robust to moderate departure from normalityPopulations have equal variances Less sensitive

27、 to this requirement when samples are of equal size from each population38Why ANOVA?Could Compare the Means One by One using Z or t Tests for Difference of MeansEach Z or t Test Contains Type I ErrorThe Total Type I Error with k Pairs of Means is 1- (1 - a) kE.g. If there are 5 means and use a = .05

28、 Must perform 10 comparisonsType I Error is 1 (.95) 10 = .4040% of the time you will reject the null hypothesis of equal means in favor of the alternative when the null is true!39Hypotheses of One-Way ANOVA All population means are equal No treatment effect (no variation in means among groups) At le

29、ast one population mean is different (others may be the same!) There is a treatment effect Does not mean that all population means are different40One-way ANOVA (No Treatment Effect)The Null Hypothesis is True41One-way ANOVA (Treatment Effect Present)The Null Hypothesis is NOT True42One-way ANOVA(Par

30、tition of Total Variation)Variation Due to Treatment SSAVariation Due to Random Sampling SSWTotal Variation SSTCommonly referred to as:Within Group VariationSum of Squares WithinSum of Squares ErrorSum of Squares UnexplainedCommonly referred to as:Among Group Variation Sum of Squares AmongSum of Squ

31、ares BetweenSum of Squares ModelSum of Squares ExplainedSum of Squares Treatment=+43Total Variation44Total Variation(continued)Response, XGroup 1Group 2Group 345Among-Group VariationVariation Due to Differences Among Groups.46Among-Group Variation(continued)Response, XGroup 1Group 2Group 347Summing

32、the variation within each group and then adding over all groups.Within-Group Variation48Within-Group Variation(continued)Response, XGroup 1Group 2Group 349Within-Group Variation(continued)For c = 2, this is the pooled-variance in the t-Test.If more than 2 groups, use F Test.For 2 groups, use t-Test.

33、 F Test more limited.50One-way ANOVAF Test StatisticTest Statistic MSA is mean squares amongMSW is mean squares withinDegrees of Freedom 51One-way ANOVA Summary TableSource ofVariationDegrees of FreedomSum ofSquaresMean Squares(Variance)FStatisticAmong(Factor)c 1SSAMSA = SSA/(c 1 )MSA/MSWWithin(Erro

34、r)n cSSWMSW =SSW/(n c )Totaln 1SST =SSA + SSW52Features of One-way ANOVA F StatisticThe F Statistic is the Ratio of the Among Estimate of Variance and the Within Estimate of Variance The ratio must always be positive df1 = c -1 will typically be small df2 = n - c will typically be large The Ratio Sh

35、ould be Close to 1 if the Null is True53Features of One-way ANOVA F StatisticIf the Null Hypothesis is FalseThe numerator should be greater than the denominatorThe ratio should be larger than 1(continued)54One-way ANOVA F Test ExampleAs production manager, you want to see if 3 filling machines have

36、different mean filling times. You assign 15 similarly trained and experienced workers, 5 per machine, to the machines. At the .05 significance level, is there a difference in mean filling times?Machine1 Machine2 Machine325.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60

37、20.4055One-way ANOVA Example: Scatter Diagram272625242322212019Time in SecondsMachine1 Machine2 Machine325.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.4056One-way ANOVA Example ComputationsMachine1 Machine2 Machine325.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.

38、7523.74 22.75 20.6025.10 21.60 20.4057Summary TableSource ofVariationDegrees of FreedomSum ofSquaresMean Squares(Variance)FStatisticAmong(Factor)3-1=247.164023.5820MSA/MSW=25.60Within(Error)15-3=1211.0532.9211Total15-1=1458.217258One-way ANOVA Example SolutionF03.89H0: 1 = 2 = 3H1: Not All Equal = .

39、05df1= 2 df2 = 12 Critical Value(s):Test Statistic: Decision:Conclusion:Reject at = 0.05There is evidence that at least one i differs from the rest. = 0.05FMSAMSW2358209211256.59Solution In EXCELUse Tools | Data Analysis | ANOVA: Single Factor EXCEL Worksheet that Performs the One-way ANOVA of the example60The Tukey-Kramer ProcedureTells which Population