Analysis of Variance Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Assumptions The replicates are obtained independently and randomly from each of the populations. The replicates from each population follow a (approximate) normal distribution. The normal populations all have a common variance. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Measuring Variation SS(factor) measures between-sample variation [SS(between)] SS(error) measures within-sample variation [SS(within)] SS(total) measures the total variation in the sample [SS(factor)] [SS(error)] Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Determining Sum of Squares Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Anova Table Source df SS MS F Factor k - 1SS(factor)MS(factor)MS(factor) Error n - 2SS(error)MS(error)MS(error) Total n - 1SS(total) MS(factor)  SS(factor) k  1 MS(factor) MS(error)  SS(error) n  k F = Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Test for Equal Variances H o :  1 2 =  2 2 = …. =  k 2 H a : At least 2 variances are unequal Reject H O if H > H Table A.14 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Confidence Intervals in One Factor ANOVA Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Multiple Comparisons Procedure Find Q , k,  using Table A.16 Determine Place the sample means in order, from smallest to largest. If two means differ by more than D, the conclusion is that the corresponding population means are unequal. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

One-Factor ANOVA Procedure Requirements The replicates are obtained independently and randomly from each of the populations. The observations from each population follow (approximately) a normal distribution. The populations all have a common variance. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

H o  1 =  2 = … =  k H a not all  ’s are equal Source df SS MS F Factor k - 1SS(factor)MS(factor)MS(factor) Error n - 2SS(error)MS(error)MS(error) Total n - 1SS(total) Reject H o if F * > F , k-1,n-1 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Randomized Block Design Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing The samples are not independent, the data are grouped (blocked) by another variable. The difference between the randomized block design and the completely randomized design is that here we use a blocking strategy rather than independent samples to obtain a more precise test for examining differences in the factor level means.

Randomized Block Design k = number of factor levels in the design b = number of blocks in the design n = number of observations = bk Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Factor Hypothesis Test H o  1 =  2 = … =  k H a not all  ’s are equal Reject H o if F * > F , k-1, (k-1)(b-1) Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Block Hypothesis Test H o  1 =  2 = … =  b H a not all  ’s are equal Reject H O if F * > F , b-1, (k-1)(b-1) Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Confidence Interval Difference Between Two Means Randomized Block (1-  ) 100% Confidence Interval Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Multiple Comparisons Procedure: Randomized Block Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing | X i  X j | > D

Two-Way Factorial Design Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Hypothesis Test Factor A H o Factor A is not significant H a Factor A is significant Reject H o,A if F 1 > F ,  1,  2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Hypothesis Test Factor B H o Factor B is not significant H a Factor B is significant Reject H o,B if F 2 > F ,  1,  2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Hypothesis Test Interaction H o Interaction is not significant H a Interaction is significant Reject H o,AB if F 2 > F ,  1,  2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Multiple Comparisons Procedure: Two-Way Factorial Design Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing