# sol3 - MIT OpenCourseWare http/ocw.mit.edu 2.830J 6.780J...

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Unformatted text preview: 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 . MIT 2.830/6.780 Problem Set 3 (2008) — Solutions Part 1 Histograms and normal probability plots for intermingled samples taken from two populations, x 1 ~ N(0,1) and x 2 ~ N( d ,1), for values of d between 0 and 4: Histogram for d = 0 Normal probability plot for d = 0 300 0.999 0.997 0.99 0.98 Frequency Frequency Frequency Frequency Frequency Frequency Frequency Frequency Frequency Frequency Frequency Frequency 0.95 0.90 200 100 0 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001 −4 −2 0 2 4 −2 −1 1 2 3 Value of parameter Value of parameter Histogram for d = 0.5 Normal probability plot for d = 0.5 300 0.999 0.997 0.99 0.98 0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001 200 100 0 −4 −2 2 4 −2 0 2 Value of parameter Value of parameter Histogram for d = 1 Normal probability plot for d = 1 300 0.999 0.997 0.99 0.98 0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001 200 100 0 −4 −2 2 4 −2 0 2 Value of parameter Value of parameter Histogram for d = 1.5 Normal probability plot for d = 1.5 300 0.999 0.997 0.99 0.98 0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001 200 100 0 −4 −2 2 4 6 −2 0 2 4 Value of parameter Value of parameter Histogram for d = 2 Normal probability plot for d = 2 200 0.999 0.997 0.99 0.98 0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001 100 0 −4 −2 2 4 6 −2 0 2 4 Value of parameter Value of parameter Histogram for d = 4 Normal probability plot for d = 4 200 0.999 0.997 0.99 0.98 0.95 0.90 0.75 100 0 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001 −5 0 5 10 −2 2 4 6 Value of parameter Value of parameter Simply looking at these normal probability plots, one would probably conclude that the distributions underlying the samples for values of d up to and including 2 could be reasonably approximated by a normal distribution. Only for the case d =4 is the sample clearly from a non-normal distribution. We might use various tests of normality to probe further. For this particular set of samples, the Lilliefors test rejects the hypothesis of normality at the 5% level for the cases d =2 and d =4. However, repeating the random sampling operation a few times shows that this is not always the result: depending on the samples that happen to be generated, the hypothesis of normality is sometimes rejected for d = 1 and d = 1.5. So while normal probability plots and tests of normality are useful in deciding whether or not we can approximate a particular distribution as normal — in order, for example, to allow further hypothesis testing — they cannot be relied upon to alert us to features of the data that we had already inadvertently ignored....
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sol3 - MIT OpenCourseWare http/ocw.mit.edu 2.830J 6.780J...

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