ANOVA0-2 - 4/24/11 AnalysisofVarianceTests...

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 4/24/11 ANOVA Dependent variable:  The variable for which a value  is measured or observed. It is a quantitative variable. Independent variable:   A variable that is observed or controlled for the purpose of determining its effect on  the value of the dependent variable. It can be  qualitative or quantitative. An independent variable is called  factor, The experiment may involve different  factor levels , Purpose:  Examines two or more levels of an  independent variable to determine if their population  means could be equal.
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 4/24/11 Examples An experiment to study the effects of 5 different  brands of gasoline on automobile engine  operating efficiency (mpg). An experiment to study the effects of presence of  4 different sugar solutions(glucose, sucrose,  fructose, mixture of 3) on bacterial growth. An experiment to investigate whether hardwood  concentration in pulp (%) has an effect on tensile  strength of bags made from pulp. An experiment to decide whether the color 
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 4/24/11 One-Way ANOVA Hypotheses: H0:  µ 1 =  µ 2 = . .. =  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
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 4/24/11
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 4/24/11 ANOVA Model: Assumption:  The samples have been independently selected, The population (treatment) variances are  equal ( 1 =  2 = … =  t ) σ σ σ The population distribution is normal Hypotheses  (alternative representation):    H0:  1 =   2 = . .. =    τ τ τ t  = 0  ij j ij x ε τ μ + + =
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 4/24/11 One-Way ANOVA, cont. Format for data: Group 1 2 J t X11 X12 X1j X1t X21 X22 X2j X2t Xi1 Xi2 Xij Xit Xn1 Xn2 Xnj Xnt X1-bar X2-bar Xj-bar Xt-bar n1 n2 nj nt
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 4/24/11 One-Way ANOVA, cont. Format for data:  Data appear in separate columns or  rows, organized by treatment groups.  Sample size of  each group may differ. Calculations: SST = SSTR + SSE 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) . 2 ) ( SS T ∑∑ = x i j x x
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 4/24/11 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) .
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ANOVA0-2 - 4/24/11 AnalysisofVarianceTests...

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