CHAPTER 12 Analysis of variance tests to accompany Introduction to business statistics fourth edition by ronald M. Weiers Presentation by priscilla chaffe-Stengel Donald n. Stengel o The Wadsworth Group
CHAPTER 12 Analysis of Variance Tests to accompany Introduction to Business Statistics fourth edition, by Ronald M. Weiers Presentation by Priscilla Chaffe-Stengel Donald N. Stengel © 2002 The Wadsworth Group
l Chapter 12- Learning objectives Describe the relationship between analysis of variance, the design of experiments, and the types of applications to which the experiments are applied Differentiate one-way, randomized block, and two-way analysis of variance techniques Arrange data into a format that facilitates their analysis by the appropriate analysis of variance technique Use the appropriate methods in testing ypotheses relative to the experimental data o 2002 The Wadsworth Group
Chapter 12 - Learning Objectives • Describe the relationship between analysis of variance, the design of experiments, and the types of applications to which the experiments are applied. • Differentiate one-way, randomized block, and two-way analysis of variance techniques. • Arrange data into a format that facilitates their analysis by the appropriate analysis of variance technique. • Use the appropriate methods in testing hypotheses relative to the experimental data. © 2002 The Wadsworth Group
l Chapter 12-Key terms ° Factor level, treatment,· Two-way analysis of block, interaction variance, factorial Within-group experiment variation Sum of squares ° Between-group Treatment variation Error Completely randomized design Block Interaction Randomized block design Total o 2002 The Wadsworth Group
Chapter 12 - Key Terms • Factor level, treatment, block, interaction • Within-group variation • Between-group variation • Completely randomized design • Randomized block design • Two-way analysis of variance, factorial experiment • Sum of squares: – Treatment – Error – Block – Interaction – Total © 2002 The Wadsworth Group
l Chapter 12-Key Concepts Differences in outcomes on a dependent variable may be explained to some degree by differences in the independent variables Variation between treatment groups captures the effect of the treatment Variation within treatment groups represents random error not explained by the experimental treatments o 2002 The Wadsworth Group
Chapter 12 - Key Concepts • Differences in outcomes on a dependent variable may be explained to some degree by differences in the independent variables. • Variation between treatment groups captures the effect of the treatment. Variation within treatment groups represents random error not explained by the experimental treatments. © 2002 The Wadsworth Group
ll One-Way anovA o Purpose: Examines two or more levels of an independent variable to determine if their population means could be equal ypotheses: 0·1 H: At least one of the treatment group means differs from the rest, or At least two of the population means are not equal where t= number of treatment groups or levels o 2002 The Wadsworth Group
One-Way ANOVA • Purpose: Examines two or more levels of an independent variable to determine if their population means could be equal. • Hypotheses: – H0 : µ1 = µ2 = ... = µt * – H1 : At least one of the treatment group means differs from the rest. OR At least two of the population means are not equal. * where t = number of treatment groups or levels © 2002 The Wadsworth Group
llOne-Way AnovA, cont Format for data: Data appear in separate columns or rows, organized as treatment groups. Sample size of each group may differ Calculations SST= SSTR SSE definitions follow) Sum of squares total sst)= sum of squared differences between each individual data value (regardless of group membership) minus the grand mean, x, across all data. total variation in the data not variance SST=∑∑(x1,-x)2 o 2002 The Wadsworth Group
One-Way ANOVA, cont. • Format for data: Data appear in separate columns or rows, organized as treatment groups. Sample size of each group may differ. • Calculations: – SST = SSTR + SSE (definitions follow) – Sum of squares total (SST) = sum of squared differences between each individual data value (regardless of group membership) minus the grand mean, , across all data... total variation in the data (not variance). SST = ( – x)2 ij x x © 2002 The Wadsworth Group
ll One-Way AnovA,cont Calculations, cont Sum of squares treatment (SSTr)=sum of squared differences between each group mean and the grand mean, balanced by sample size.between groups variation (not variance) SSTR=∑n.(x Sum of squares error sse)=sum of squared differences between the individual data values and the mean for the group to which each belongs.within group variation(not variance) SSE=∑∑(x1;-x o 2002 The Wadsworth Group
One-Way ANOVA, cont. • Calculations, cont.: – Sum of squares treatment (SSTR) = sum of squared differences between each group mean and the grand mean, balanced by sample size... betweengroups variation (not variance). – Sum of squares error (SSE) = sum of squared differences between the individual data values and the mean for the group to which each belongs... withingroup variation (not variance). SSTR ( – x)2 j x j = n SSE = (x ij – x j )2 © 2002 The Wadsworth Group
lLOne-Way ANOVA, cont Calculations, cont Mean square treatment(MSTR)=SstR/(t-1) where t is the number of treatment groups. between groups varlance Mean square error(Mse)=SSE/(N-t)where n is the number of elements sampled and t is the number of treatment groups. within-groups variance F-Ratio= MStR/MSE, where numerator degrees of freedom are t-1 and denominator degrees of freedom are N-t o 2002 The Wadsworth Group
One-Way ANOVA, cont. • Calculations, cont.: – Mean square treatment (MSTR) = SSTR/(t – 1) where t is the number of treatment groups... betweengroups variance. – Mean square error (MSE) = SSE/(N – t) where N is the number of elements sampled and t is the number of treatment groups... within-groups variance. – F-Ratio = MSTR/MSE, where numerator degrees of freedom are t – 1 and denominator degrees of freedom are N – t. © 2002 The Wadsworth Group
l One-Way ANoVA- An example if occupancy of a vehicle might be related to the speed r da Problem 12.30: Safety researchers, interested in determinin which the vehicle is driven, have checked the following speed(MPH)measurements for two random samples of vehicles Driver alone. 64507155676180565974 1+ rider(s:445254486967545758516267 a. What are the null and alternative hypotheses? 11=12 where group 1= driver alone H1:A1≠A2 Group 2= with rider(s) o 2002 The Wadsworth Group
One-Way ANOVA - An Example Problem 12.30: Safety researchers, interested in determining if occupancy of a vehicle might be related to the speed at which the vehicle is driven, have checked the following speed (MPH) measurements for two random samples of vehicles: Driver alone: 64 50 71 55 67 61 80 56 59 74 1+ rider(s): 44 52 54 48 69 67 54 57 58 51 62 67 a. What are the null and alternative hypotheses? H0 : µ1 = µ2 where Group 1 = driver alone H1 : µ1 µ2 Group 2 = with rider(s) © 2002 The Wadsworth Group
ul One-Way aNOva- An example b Use ANOVA and the 0.025 level of significance in testing the appropriate null hypothesis 1=637,S1=9.357 56.916.S~=7.806.n、=12 x=60.0 SSTR=10(637-602+12(56917-602=250983 SSE=(64-637)2+(50-63.7)2+…+(74-63.7)2 +(44-56917)2+(52-56917)2+…+(67-56917)2 1487.017 SSTotal=(64-60)2+(50-60)2+…+(74-60)2 +(44-60)2+(52-60)2+…+(67-60)2 1738 o 2002 The Wadsworth Group
One-Way ANOVA - An Example b. Use ANOVA and the 0.025 level of significance in testing the appropriate null hypothesis. SSTR = 10(63.7 – 60)2 + 12(56.917 – 60)2 = 250.983 SSE = (64 – 63.7 )2 + (50 – 63.7 )2 + ... + (74 – 63.7 )2 + (44 – 56.917) 2 + (52 – 56.917) 2 + ... + (67 – 56.917) 2 = 1487.017 SSTotal = (64 – 60 )2 + (50 – 60 )2 + ... + (74 – 60 )2 + (44 – 60) 2 + (52 – 60) 2 + ... + (67 – 60) 2 = 1738 x 1 = 63.7, s 1 = 9.3577, n 1 = 10 x 2 = 56.916 , s 2 = 7.806, n 2 = 12 x = 60.0 © 2002 The Wadsworth Group
