q2_sol_2008

q2_sol_2008 - MIT OpenCourseWare http:/ocw.mit.edu 2.830J /...

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MIT OpenCourseWare http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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Name: ______SOLUTIONS____________________________ Mean: 71.6%; standard deviation 15.3%. 28 students sat the exam. Massachusetts Institute of Technology Department of Mechanical Engineering Department of Electrical Engineering and Computer Science 2.830J/6.780J Control of Manufacturing Processes Spring 2008 Quiz #2 Thursday – April 24, 2008 In all problems, please show your work and explain your reasoning. Statistical tables for the cumulative standard normal distribution, percentage points of the χ 2 distribution, percentage points of the t distribution, and percentage points of the F distribution (all from Montgomery, 5 th Ed.) are provided. Problem 1 [45%] An experiment is designed and executed, in which the design or input variable is x and the output variable is y. The input range is normalized to [-1, +1]. Experiments are run in the order shown in Table 1 below, with the input setting and output result as given in the table. Table 1: Full factorial DOE experiment results Run # xy 1 -1 8 2 1 18 3 -1 9 4 1 19 5 -1 10 6 1 20 Part (a) [5%] Fit a model of the form y = β 0 + β 1 x to the data, and determine point estimates for β 0 and β 1 . ANSWER: Since the design is normalized to ±1 and is balanced, we can use simplified contrasts for the estimation of the offset and linear terms: β 0 = overall average = 14 β 1 = (19-9)/2 = 5 Or y = 14 + 5 x 1
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Part (b) [10%] Determine the standard error (std. err.) and 95% confidence intervals for the estimates of β 0 and β 1 . Are both parameters significant to 95% confidence or better? Should you include both terms in the model? ANSWER: First, we need an estimate of the underlying pure error. The residuals for runs 1 through 6 are -1, -1, 0, 0, +1, +1, giving a SS E = 4, with degrees of freedom = 4. So the MS E is 4/4 = 1. That is to say, our estimate for σ 2 E is 1. With this error estimate, we can now estimate the variance for both the constant and slop terms. For the constant term, 408 . 0 6 1 ) Var( 2 0 = = = n σ β And for the slope term, 408 . 0 6 1 ) ( ) Var( 2 2 2 1 = = = = n x x i σσ . Thus the standard error for each estimate is 408 . 0 6 1 err. std. = = . Finally, we can formulate the 95% confidence intervals using the t distribution with 4 degrees of freedom (4 since we have a pooled estimate of variance using two different x levels): 776 . 2 4 , 025 . 0 , 2 / = = t t να , so that the confidence intervals are at 13 . 1 ) 408 . 0 ( 776 . 2 ± = ± around the point estimates, i.e.: 13 . 1 14 0 ± = or 13 . 15 87 . 12 0 and 13 . 1 5 1 ± = or 13 . 6 87 . 3 0 . Part (c) [5%] Following good practice, we next examine the residuals (differences between the model prediction values and measured values, for our data). In particular, we consider the residuals as a function of run order. What pattern in the residuals raises a concern? What modifications might you suggest to the experimental design or analysis in light of this?
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This note was uploaded on 09/24/2010 for the course MECHE 2.830J taught by Professor Davidhardt during the Spring '08 term at MIT.

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q2_sol_2008 - MIT OpenCourseWare http:/ocw.mit.edu 2.830J /...

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