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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 16

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online