Chapter 8 Student Lecture Notes 8-1. Department of Quantitative Methods & Information Systems. Business Statistics

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1 Chapter 8 Student Lecture Notes 8-1 Department of Quantitative Methods & Information Systems Business Statistics Chapter 11 One Way analysis of Variance QMIS 0 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should be able to: Recognize situations in which to use analysis of variance Perform a single-factor hypothesis test and interpret results QMIS 0, by Dr. M. Zainal

2 Chapter 8 Student Lecture Notes 8- Chapter Overview Analysis of Variance (ANOVA) One-Way ANOVA F-test Tukey- Kramer test Randomized Complete Block ANOVA F-test Fisher s Least Significant Difference test Two-factor ANOVA with replication General ANOVA Setting Investigator controls one or more independent variables Called factors (or treatment variables) Each factor contains two or more levels (or categories/classifications) Observe effects on dependent variable Response to levels of independent variable Experimental design: the plan used to test hypothesis QMIS 0, by Dr. M. Zainal

3 Chapter 8 Student Lecture Notes 8-3 One-Way Analysis of Variance Evaluate the difference among the means of three or more populations Examples: Accident rates for 1 st, nd, and 3 rd shift Expected mileage for five brands of tires Assumptions Populations are normally distributed Populations have equal variances Samples are randomly and independently drawn Completely Randomized Design Experimental units (subjects) are assigned randomly to treatments Only one factor or independent variable With two or more treatment levels Analyzed by One-factor analysis of variance (one-way ANOVA) Called a Balanced Design if all factor levels have equal sample size QMIS 0, by Dr. M. Zainal

4 Chapter 8 Student Lecture Notes 8-4 Hypotheses of One-Way ANOVA H0 :μ1 μ μ3 μk All population means are equal i.e., no treatment effect (no variation in means among groups) H A :Not allof the populationmeansare 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) One-Factor ANOVA H H 0 :μ1 μ μ3 μk A :Not allμ i are the same All Means are the same: The Null Hypothesis is True (No Treatment Effect) μ 1 μ μ3 QMIS 0, by Dr. M. Zainal

5 Chapter 8 Student Lecture Notes 8-5 One-Factor ANOVA H H 0 :μ1 μ μ3 μk A :Not allμ i are the same (continued) At least one mean is different: The Null Hypothesis is NOT true (Treatment Effect is present) or μ 1 μ μ3 μ1 μ μ3 Partitioning the Variation Total variation can be split into two parts: SST = SSB + SSW SST = Total Sum of Squares SSB = Sum of Squares Between SSW = Sum of Squares Within QMIS 0, by Dr. M. Zainal

6 Chapter 8 Student Lecture Notes 8-6 Partitioning the Variation SST = SSB + SSW (continued) Total Variation (SST) = the aggregate dispersion of the individual data values across the various factor levels Between-Sample Variation (SSB) = dispersion among the factor sample means Within-Sample Variation (SSW) = dispersion that exists among the data values within a particular factor level Partition of Total Variation Total Variation (SST) Variation Due to Factor (SSB) = + Commonly referred to as: Sum of Squares Between Sum of Squares Among Sum of Squares Explained Among Groups Variation Variation Due to Random Sampling (SSW) Commonly referred to as: Sum of Squares Within Sum of Squares Error Sum of Squares Unexplained Within Groups Variation QMIS 0, by Dr. M. Zainal

7 Chapter 8 Student Lecture Notes 8-7 Total Sum of Squares SST = SSB + SSW SST k n i i 1 j 1 ( x ij x) Where: SST = Total sum of squares k = number of populations (levels or treatments) n i = sample size from population i x ij = j th measurement from population i x = grand mean (mean of all data values) Total Variation (continued) SST (x 11 x) (x 1 x)... (x kn x) k Response, X x Group 1 Group Group 3 QMIS 0, by Dr. M. Zainal

8 Chapter 8 Student Lecture Notes 8-8 Where: Sum of Squares Between SST = SSB + SSW SSB n i(x k i 1 x) SSB = Sum of squares between k = number of populations n i = sample size from population i x i = sample mean from population i x = grand mean (mean of all data values) i Between-Group Variation SSB n i(x k i 1 x) Variation Due to Differences Among Groups i SSB MSB k 1 Mean Square Between = SSB/degrees of freedom i j QMIS 0, by Dr. M. Zainal

9 Chapter 8 Student Lecture Notes 8-9 Between-Group Variation (continued) SSB n (x 1 1 x) n(x x)... nk(xk x) Response, X x x1 x 3 x Group 1 Group Group 3 Sum of Squares Within SST = SSB + SSW k nj SSW (xij xi) i 1 j 1 Where: SSW = Sum of squares within k = number of populations n i = sample size from population i x i = sample mean from population i x ij = j th measurement from population i QMIS 0, by Dr. M. Zainal

10 Chapter 8 Student Lecture Notes 8-10 Within-Group Variation SSW k i 1 j 1 (x x Summing the variation within each group and then adding over all groups n j ij i ) MSW SSW n k T Mean Square Within = SSW/degrees of freedom i Within-Group Variation (continued) SSW (x 11 x1) (x1 x)... (xkn k xk ) Response, X x 3 x 1 x Group 1 Group Group 3 QMIS 0, by Dr. M. Zainal

11 Chapter 8 Student Lecture Notes 8-11 One-Way ANOVA Table Source of Variation Between Samples Within Samples SS df MS SSB SSB k - 1 MSB = k - 1 SSW SSW n T - k MSW = n T - k Total SST = n T - 1 SSB+SSW F ratio MSB F = MSW k = number of populations n T = sum of the sample sizes from all populations df = degrees of freedom Formula can be used SSB(SSF) SSB = SST-SSW SSB = MSB (k-1) SSB = T 1 SSB = + T n T k n n k n i x i x T n SSW(SSE) SSW = SST-SSB SSW = MSW (n-k) SSW = n 1 1 S 1 + n 1 S + + n k 1 S k = (n i 1)S i SSW= x T 1 + T n 1 n + + T k n k SST SST = SSB+SSW SST = x T n QMIS 0, by Dr. M. Zainal QMIS 0, by Dr. M. Zainal

12 Chapter 8 Student Lecture Notes 8-1 Notes k: number of samples/ groups/ populations. n = n 1 + n + + n k (total sample size). T 1 = x 1, T = x,, T k = x k or T 1 = n 1 x 1, T = n x,, T k = n k x k T = T 1 + T + + T k S 1, S,, S k x = x 1 + x + + x k x x T i x i T i = x i MSE = S P = n 1 1 S 1 + n 1 S + + n k 1 S k n k QMIS 0, by Dr. M. Zainal 3 One-Factor ANOVA F Test Statistic H 0 : μ 1 = μ = = μ k H A : At least two population means are different Test statistic F MSB MSW Degrees of freedom MSB is mean squares between variances MSW is mean squares within variances df 1 = k 1 (k = number of populations) df = n T k (n T = sum of sample sizes from all populations) QMIS 0, by Dr. M. Zainal

13 Chapter 8 Student Lecture Notes 8-13 Interpreting One-Factor ANOVA F Statistic The F statistic is the ratio of the between estimate of variance and the within estimate of variance The ratio must always be positive df 1 = k -1 will typically be small df = n T - k will typically be large The ratio should be close to 1 if H 0 : μ 1 = μ = = μ k is true The ratio will be larger than 1 if H 0 : μ 1 = μ = = μ k is false One-Factor ANOVA F Test Example You want to see if three different golf clubs yield different distances. You randomly select five measurements from trials on an automated driving machine for each club. At the.05 significance level, is there a difference in mean distance? Club 1 Club Club QMIS 0, by Dr. M. Zainal

14 Chapter 8 Student Lecture Notes 8-14 One-Factor ANOVA Example: Scatter Diagram 1 3 Club One-Factor ANOVA Example Computations Club 1 Club Club QMIS 0, by Dr. M. Zainal

15 Chapter 8 Student Lecture Notes 8-15 One-Factor ANOVA Example Solution SUMMARY Groups Count Sum Average Variance Club Club Club ANOVA Source of Variation Between Groups Within Groups ANOVA -- Single Factor: Excel Output EXCEL: tools data analysis ANOVA: single factor SS df MS F P-value F crit E Total QMIS 0, by Dr. M. Zainal

16 Chapter 8 Student Lecture Notes 8-16 Chapter Summary Described one-way analysis of variance The logic of ANOVA ANOVA assumptions F test for difference in k means Copyright The materials of this presentation were mostly taken from the PowerPoint files accompanied Business Statistics: A Decision-Making Approach, 7e 008 Prentice-Hall, Inc. QMIS 0, by Dr. M. Zainal Chap 10-3 QMIS 0, by Dr. M. Zainal