同济大学：《应用统计学》课程电子教案（课件讲稿）Chapter 10 Analysis of Variance（ANOVA）

Statistics for Managers Using Microsoft Excel Chapter 10 Analysis of Variance (ANOVA)

Chapter 10 Analysis of Variance (ANOVA) Statistics for Managers Using Microsoft Excel

Chapter Goals After completing this chapter,you should be able to: Recognize situations in which to use analysis of variance Understand different analysis of variance designs Perform a single-factor hypothesis test and interpret results Conduct and interpret post-hoc multiple comparisons procedures Analyze two-factor analysis of variance tests

Chapter Goals After completing this chapter, you should be able to:  Recognize situations in which to use analysis of variance  Understand different analysis of variance designs  Perform a single-factor hypothesis test and interpret results  Conduct and interpret post-hoc multiple comparisons procedures  Analyze two-factor analysis of variance tests

-Is there any difference among suppliers? If there is,which supplier's product is best? Supplier 1 Supplier 2 Supplier 3 Supplier 4 18.5 26.3 20.6 25.4 24.0 25.3 25.2 19.9 17.2 24.0 20.8 22.6 19.9 21.2 24.7 17.5 18.0 24.5 22.9 20.4

 Is there any difference among suppliers?  If there is, which supplier’s product is best? Supplier 1 Supplier 2 Supplier 3 Supplier 4 18.5 26.3 20.6 25.4 24.0 25.3 25.2 19.9 17.2 24.0 20.8 22.6 19.9 21.2 24.7 17.5 18.0 24.5 22.9 20.4

Chapter Overview Analysis of Variance(ANOVA) Data Analysis Analysis Tools OK Anova:Single Factor Anova:Two-Factor With Replication Cancel Anova:Two-Factor Without Replication Correlation Covariance Help Descriptive Statistics Exponential Smoothing F-Test Two-Sample for Variances Fourier Analysis Histogram

Chapter Overview Analysis of Variance (ANOVA) F-test Tukey￾Kramer test One-Way ANOVA Two-Way ANOVA Interaction Effects

General ANOVA Setting ◆ Investigator controls one or more independent variables Called factors (or treatment variables) Each factor contains two or more levels (or groups or categories/classifications) Observe effects on the dependent variable Response to levels of independent variable Experimental design:the plan used to collect the data

General ANOVA Setting  Investigator controls one or more independent variables  Called factors (or treatment variables )  Each factor contains two or more levels (or groups or categories/classifications)  Observe effects on the dependent variable  Response to levels of independent variable  Experimental design: the plan used to collect the data

One-Way Analysis of Variance Experimental units(subjects)are assigned randomly to treatments Subjects are assumed homogeneous Only one factor or independent variable With two or more treatment levels Analyzed by one-factor analysis of variance (one-way ANOVA)

One-Way Analysis of Variance  Experimental units (subjects) are assigned randomly to treatments  Subjects are assumed homogeneous  Only one factor or independent variable  With two or more treatment levels  Analyzed by one-factor analysis of variance (one-way ANOV A )

One-Way Analysis of Variance Evaluate the difference among the means of three or more groups Examples:Accident rates for 1st,2nd,and 3rd shift Expected mileage for five brands of tires ■Assumptions Populations are normally distributed Populations have equal variances Samples are randomly and independently drawn

One-Way Analysis of Variance  Evaluate the difference among the means of three or more groups Examples: Accident rates for 1st, 2nd, and 3rd shift Expected mileage for five brands of tires  Assumptions  Populations are normally distributed  Populations have equal variances  Samples are randomly and independently drawn

xii X~N(4,c2）i=1,2,,a;j=1,2,,n E≈N0,σ2) Level Ar A2 Aa Observations X11 X21 Xal 2 Xaj XInl X2n2 Xana Population 41 42 a mean

Level A1 A2 ··· Aa Observations x11 x21 ··· xa1 x1j x2j ··· xaj x1n1 x2n2 ··· xana Population mean 1 2 a Xi ～N( i,  2 ) xij = i + ij i = 1, 2,···, a；j = 1, 2,···, ni ij ～ N(0, 2 )

Hypotheses of One-Way ANOVA ■H01=2=μ3=…=c All population means are equal i.e.,no treatment effect(no variation in means among groups) H:Not all of the population means are the same At least one population mean is different i.e.,there is a treatment effect Does not mean that all population means are different (some pairs may be the same)

Hypotheses of One-Way ANOVA   All population means are equal  i.e., no treatment effect (no variation in means among groups)   At least one population mean is different  i.e., there is a treatment effect  Does not mean that all population means are different (some pairs may be the same) 0 μ1 μ 2 μ 3 μ c H :      H1 : Not all of the population means are the same

One-Factor ANOVA H0μ1=2=μ3=…=c H,Not all u are the same All Means are the same: The Null Hypothesis is True (No Treatment Effect) 1=2=3

One-Factor ANOVA All Means are the same: The Null Hypothesis is True (No Treatment Effect) 0 μ1 μ 2 μ 3 μ c H :      H1 : Not all μi are the same μ1  μ 2  μ 3