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Unformatted text preview: McGill University Advanced Business Statistics MGSC272 Fall07 Analysis of Variance Read: Business Statistics (A Second Course) , Custom Edition for McGill University Chapter 10 ANOVA ANOVA is a statistical test of significance for the equality of several (2 or more) population (treatment) means. Assumptions: 1. All populations are normally distributed. 2. All populations have the same variance: 1 2 = 2 2 = 3 2 = ........ = k 2 = 2 . 3. Independent random samples. Notation Y ij = the i th observation from the j th treatment. = the mean of the j th sample. T j = the total of the j th sample. n j = the size of the j th sample. n T = n j = the total number of observations. p = the number of treatments. = the overall (grand) mean. j = the mean of the j th treatment. = the common standard deviation for all treatments. j Y Y ANOVA Table Source of Sum of Variation Squares df Mean Square F* Treatment SSTR p  1 MSTR = SSTR/(p  1) MSTR/MSE Error SSE n T p MSE = SSE/(n T p) Total SSTO n T 1 ANOVA Test of Hypothesis H : 1 = 2 = 3 = ........ = p ( All the population means are equal). H 1 : Not all j are equal (At least one of the means is different). TS: F* = MSTR/MSE (or MSB/MSW). CV: F(1  ; p  1; n T p ). DR: Conclude H if F * CV and Conclude H 1 if F * > CV. The Formulas 2 SSTO = ( ) ij Y Y ( 29 2 SSTR j j n Y Y = 2 SSE = ( ) = SSTO  SSTR ij j Y Y Partition Theorem SSTO = SSTR + SSE df TO = df TR + df E Oneway ANOVA A Completely Randomized Design A completely randomized design to compare p treatment means is one in which the treatments are randomly assigned to the experimental units, or in which independent random samples are drawn from each of the p populations. EXAMPLE A student society has undertaken a survey of the cost of a night out in three different cities: Montreal, Toronto and Ottawa. The results of the survey are as follows (in dollars): Montreal Toronto Ottawa 16 19 23 17 26 19 22 24 24 14 24 19 27 20 Summary Statistics Sample sizes : n 1 = 6 n 2 = 5 n 3 = 3 Means: 1 18 Y = 2 24 Y = 3 22 Y = n T = 14; Y = 294 / 14 = 21 Calculating Sums of Squares SSTR = = 102. SSE = = (16  18)2 + (17  18)2 + (22  18)2 + (14  18)2 + (19  18)2 + (20  18)2 + (19  24)2 + (26  24)2 + (24  24)2 + (24  24)2 + (27  24)2 + (23  22)2 + (19  22)2 + (24  22)2 = 94. SSTO = SSTR + SSE = 102 + 94 = 196. 2 2 2 2 ( ) 6 (18 21) 5 (24 21) 3 (22 21) j j n Y Y = + + 2 ( ) ij j Y Y ANOVA Table Source of Variation Sum of Squares df Mean Square F* Treatment 102 2 51.00 5.97 Error 94 11 8.56 Total 196 13 Note: E(MSE) = 2 MSE s 2 . The test of hypothesis H : 1 = 2 = 3 (All three population means are equal)....
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 Spring '12
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