FinalExam_2008_sol - CHEN 3010 Applied Data Analysis Fall...

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CHEN 3010 Applied Data Analysis Fall 2008 Final Examination Solutions 1) (30 pts) Two pilot plants are constructed at different company locations to produce small quantities of new product for test marketing in different regions of the country. Engineers have attempted to “close” material balances (“close” means account for all raw materials in the product streams via conservation of mass). They have obtained the following results based on 100 kg of raw materials. Investigate whether the results differ significantly between plants A and B. You must prove any assumptions you make about the variances of the two populations . Use α =0.05. Solution: First, we need to see if the variances are equal or unequal. This is based on an f-test: ݂ ଵ.ଷହ ଵ.ଶଵ ൌ 1.245 where the hypotheses are: ܪ ൌߪ and ܪ ൐ߪ . The critical f-value is: ݂ ଴.଴ହ,ଽ,ଽ ൌ3 .18 Since 1.245 < 3.18, we reject the alternate hypothesis and assume equal variances (and we would need to show that our test statistic is greater than 3.18 to accept the alternate). Our hypotheses are: ܪ ൌߤ ߤ ൐ߤ . For variances unknown but equal and equal sample size, we use the following equation for our test statistic: 16 . 3 2 0 = = n S X X T p B A where 2 s 2 2 A B p s S + = The critical t-value is ݐ ଴.଴ହ,ଵ଼ ൌ 1.734 . Since 3.16 > 1.734, we accept the alternate hypothesis and claim that the mean from plant A is greater than that for plant B . Plant A Plant B 97.8 97.2 98.9 100.5 101.2 98.2 98.8 98.3 102 97.5 99 99.9 99.1 97.9 100.8 96.8 100.9 97.4 100.5 97.2 AB x 99.90 x 98.09 s 1.35 s 1.21 = = ==
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CHEN 3010 Final Exam Solutions Page 2 2) (15 pts) Construct a fractional factorial design for six factors that requires only 8 experiments. Present the design in table form including a column showing the treatment combinations. Describe, in general terms, the aliasing inherent to this design. Solution: We have 6 factors but only 8 experiments. Thus, this will be a 2 6-3 design. For a 2 6-3 design, the design generators are D=AB, E=AC, and F=BC. We start by writing out the normal 23 design (columns A through C) and then add in three columns for D, E, and F, multiplying the corresponding A Æ C coded variables appropriately. This is shown below with the treatment combinations: The general aliasing relationship, since this is a resolution III design, will be that main effects will be aliased with binary and higher interactions and binary effects will be aliased with main, binary, and higher interactions.
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FinalExam_2008_sol - CHEN 3010 Applied Data Analysis Fall...

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