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Unformatted text preview: MAT 3378 (Winter 2010) Final Examination  Solutions Spring 2010 Professor G. Lamothe Duration: 3 hours Student Number: Last Name: First Name: This is a closed book examination. Only nonprogrammable and nongraphic calculators are per mitted. Two sheets double sided are permitted. There are six questions. 1 1. [10 points] A canning plant uses a large number of machines for the filling process. Each machine is supposed to pour a specified weight of product into each container. The plant manager suspects that there is too much variation in weight of the product poured among the ma chines. To check this suspicion, the plant manager selects four machines at random and weighs the contents of five randomly selected contain ers filled by each of the four randomly selected machines. a) Write down the corresponding onefactor ANOVA model. b) Complete the following ANOVA table to test for machine effects. Give the null and alternative hypotheses and the value of the corre sponding test statistic. source df SS MS F pvalue machine 3 0.0015650 < . 0001 error 0.0012 total 19 c) What are the appropriate procedures to describe the effects in the context of this problem? Note: Describe the techniques that should be used to describe the effects. You do not need to describe the effects. d) How much of the variance in the weights is attributed to differ ences among the machines? Solution: a) The appropriate model is a one factor ANOVA model with random effects: Y ij = i + ij , where i are independent N ( , 2 ); ij are independent N ( , 2 ); i , ij are independent for i = 1 ,... 4 and j = 1 ,..., 5. b) 2 source df SS MS F pvalue machine 3 .00465 0.0015650 20.87 < . 0001 error 16 0.0012 total 19 .005895 The hypotheses for testing for treatment effects are H : 2 = 0 against H a : 2 6 = 0 . The observed value of the test statistic is F * = MSTR / MSE = 20 . 87. c) Estimate the variance of the random effect 2 , but more impor tantly estimate the intraclass correlation which gives the proportion of the variance in the response which is explained by the variance of the treatment effect. d) s 2 = (MSRT MSE) /n = 0 . 000298 and so the estimated intra class correlation is d ICC = s 2 s 2 + MSE = 79 . 9% . 3 2. [10 points] Consider the following data for a randomized complete block design study where block and treatment effects are fixed. Treatment Block j = 1 j = 2 j = 3 i = 1 11.5 11.8 12.9 i = 2 15.3 13.7 16.2 i = 3 18.3 21.3 23 Here are few corresponding sums: SSBL = 120 . 08 , SSTR = 8 . 8867 , SSTO = 135 . 7 3 X i =1 3 X j =1 ( Y i  Y )( Y j Y ) Y ij = 16 . 6211 . (a) For such an ANOVA model, we do not incorporate an interaction term ( ) ij . What problem are we going to encounter if we do add the interaction term ( ) ij into the model?...
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This note was uploaded on 12/22/2011 for the course MAT 3378 taught by Professor G.lamothe during the Spring '11 term at University of Ottawa.
 Spring '11
 G.Lamothe
 Math

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