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# sol2 - MIT OpenCourseWare http/ocw.mit.edu 2.830J 6.780J...

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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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MIT 2.830/6.780 Problem Set 2 (2008) — Solutions Problem 1 Many interesting responses were received. We can discuss on an individual basis when selecting term projects. Problem 2 Let x be the concentration of particles in particles/ft 3 : x ~ N(15.08, 0.025) Since the acceptable concentration range is 15.00 ± 0.10 particles/ft 3 , the probability that a system conforms to specifications is: P (14.9 x 15.1) = P ( x 15.10) – P ( x 14.90) = Φ ([15.10–15.08]/0.05) – Φ ([14.90–15.08]/0.05) = Φ ([15.10–15.08]/0.05) – {1 – Φ ([15.08–14.90]/0.05)} = 0.06554 – 1.59 × 10 –4 = 65.5% Problem 3 Because we know neither the population’s mean nor its standard deviation, we need to use a t-test. The sample size is 10 so the number of degrees of freedom is 10 – 1 = 9. A 99% confidence interval on the mean photoresist thickness is given by: 10 μ x + t (0.01/ 2),9 10 x t (0.01/ 2),9 s s where the sample mean is x = 13.3962 μ m and the sample variance is s 2 = 1 ( x x ) 2 = 1.5277 × 10 5 μ m 2 . 9 From tables, t 0.005,9 = 3.250. The 99% confidence interval is therefore: 13.3922 μ m μ 13.4002 μ m.
Problem 4 a) Histograms and normal probability plots for shearing data , sorted by material and end-of-sample. Histogram of sheared widths 1 Normal probability plot: width 1 15 0.99 0.98 0.95 0.90 Aluminum Steel Frequency Frequency Frequency Frequency 10 5 0.10 0.05 0.02 0.01 0 1.005 1.01 1.015 1.02 1.005 1.01 1.015 Width 1 (in) Data (in) Histogram of sheared widths 2 Normal probability plot: width 2 15 0.99 0.98 0.95 0.90 Frequency Frequency Frequency Frequency 0.75 0.50 0.25 10 5 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0 1 1.01 1.02 1.03 1.005 1.01 1.015 Width 2 (in) Data (in) Histogram of sheared widths 1 Normal probability plot: width 1 15 0.99 0.98 0.95 0.90 10 5 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0 1 1.01 1.02 1.03 1.01 1.015 1.02 1.025 Width 1 (in) Data (in) Histogram of sheared widths 2 Normal probability plot: width 2 10 1 1.01 1.02 1.03 0.99 0.98 0.95 8 0.90 6 0.75 0.50 4 0.25 0.10 2 0 0.05 0.02 0.01 1.005 1.01 1.015 1.02 Width 2 (in) Data (in)

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Skewness, kurtosis and normality test results for the two materials and two widths: Steel Aluminum Width 1 Width 2 Width 1 Width 2 Skewness –0.3568 –0.7232 0.3199 –1.1607 Kurtosis* 2.8899 3.3079 3.4622 5.8867 Lillifors test at 5% level Reject Reject Accept Accept * Defined as 3 for a normal distribution. Values less than 3 correspond to
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