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# Exam1Solutions - IE 410 Midterm Exam Answers Fall 2003...

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IE 410 Midterm Exam Answers Fall 2003 Average = 78.4 90 | 7 5 3 2 2 1 1 80 | 9 8 8 6 6 6 5 5 3 0 70 | 9 9 9 6 5 5 60 | 4 4 2 0 50 | 2 40 | 3 30 | 7

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Question 1 An experiment will be conducted to see if the priced offered by car dealers for a used car varies with the characteristics of the seller of the car. Four sellers (A, B, C, D) will be selected to solicit offers from various used car dealers. Owners A and D are male while B and C are female. Owners A and C are in their 40’s while B and D are over 65 years of age. A single used car will be taken to several different dealers. Each owner will take the car to several dealers and received a cash offer. The resulting data will be recorded in units of hundreds of dollars. Suppose based on prior information, you guess that the variance σ 2 is about 3.0 (note the units are hundreds of dollars). You would like to detect a difference in means of \$400 with probability 0.05. You will conduct your anova at α =0.05. What sample size would you recommend? We will need to be careful with our units here. n n Thus dollars of units use we If n n Thus units in Then units in If of units use will we First a n D a nD 8165 . 0 ) 4 ( 2 2 . 173 400 2 . 173 8165 . 0 ) 4 ( 2 732 . 1 4 100 \$ 732 . 1 3 100 \$ 0 . 3 100 \$ 2 2 2 2 2 2 2 = = Φ = = = Φ = = = = Φ = Φ σ σ σ σ σ The following calculation would be incorrect: Φ 2 = nD 2 /2a σ 2 = n(400) 2 /(2*4*300) = 66.67n Φ = 8.165 n n v 1 =a-1 v 2 =a(n-1) Φ β 6 3 20 2 0.15 8 3 28 2.309401 0.05 9 3 32 2.44949 0.02 Thus n is in the range 8 or 9 and N is 32 to 36. Question 2
Despite your recommendation in question 1, the experiment was run, and the data below observed (again in units of hundreds of dollars) Owner A B C D -------------------------------- 25 21 26 20 24 25 29 23 26 23 27 24 27 22 28 26 23 20 29 21 28 22 28 23 a) Construct the ANOVA table and conduct the ANOVA F-test. Do not consider residual analysis in question 2 (we will do this in question 3). Even if the residuals indicate model inadequacy, pretend as if the model is adequate. . ANOVA Table for Y by Factor Analysis of Variance ----------------------------------------------------------------------------- Source Sum of Squares Df Mean Square F-Ratio P-Value ----------------------------------------------------------------------------- Between groups 121.833 3 40.6111 13.10 0.0001 Within groups 62.0 20 3.1 ----------------------------------------------------------------------------- Total (Corr.) 183.833 23 What is the P-value of the ANOVA F-test? ___0.0001____ b) What contrast would you recommend to test for differences in cash offers between sexes? L sex = μ A + μ D - μ B - μ C The answer “age” is wrong, and makes no sense since I clearly said that I was interested in looking at differences between the sexes.

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c) Estimate the contrast in part b) and provide a 99% confidence interval. 667 . 1 ˆ - = - - + = C B D A sex Y Y Y Y L From equation 3-30 (page 92) a confidence interval may be computed for a contrast as follows: 2.423] , -5.757 [ 09 . 4 667 . 1 ) 1.4376 )( 845 . 2 ( 667 . 1 ) 4 ( 6 1 . 3 667 . 1 20 , 005 . 0 2 , 2 / . = ± - = ± - = ± - = ± - t t i i i E a N i i c n MS y c α
d) Test the contrast you recommended in b). What is the P-value of the test?__0.2722____ ( 29 1667 . 4 ) 4 ( 6 2 10 1 2 2 1 . = - = = = = a i ci n a i ciyi SSc 344 . 1 1 . 3 16667 . 4 a - N SSE 1 SSc = = Cumulative Distribution ------------------------------------ Distribution: F (variance ratio) Lower Tail Area (<) Variable Dist. 1 Dist. 2 Dist. 3 Dist. 4 Dist. 5 1.344 0.727778 Probability Density Variable Dist. 1 Dist. 2 Dist. 3 Dist. 4 Dist. 5 1.344 0.171673 Upper Tail Area (>) Variable Dist. 1 Dist. 2 Dist. 3 Dist. 4 Dist. 5 1.344 0.272222 The statgraphics anova routine states the P-value as 0.26.

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Exam1Solutions - IE 410 Midterm Exam Answers Fall 2003...

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