ISE 352: Design of Experiments
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ISE 352: Design of Experiments
• Dr. Laura Moody• Course Objectives:
– Upon successful completion of this course, you should be able to do the following:• Plan an experiment.• Identify the appropriate experimental method.• Develop a detailed experimental design.• Conduct and analyze the results of an experiment.• Draw appropriate conclusions and specific
recommendations.
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Experimentation
• What is an experiment?
• Give an example of an experiment you have conducted …
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Why experiment?
• Process improvement– improve productivity & safety– reduce variability / improve quality– reduce development time– reduce overall costs
• Engineering design– evaluation / comparison of design configurations– evaluation of alternatives– select design parameters for robust design /
performance– explore product ideasISE 352 - Ch. 1&2 3
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Basic Principles
• randomization
• replication
• blocking
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Guidelines … with an example
• Example: can you improve the range of a remote car key?1. Recognition and statement of the problem:
2. Selection of the response variable:
3. Choice of factors, levels, and range:
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Guidelines … with an example (cont.)
4. Choice of experimental design:
5. Performing the experiment:
6. Statistical analysis of the data:
7. Conclusions & recommendations:
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Some basic statistical concepts
• Describing sample data– Random samples– Sample mean, variance, standard deviation– Populations versus samples– Population mean, variance, standard deviation– Estimating parameters
• Simple comparative experiments– The hypothesis testing framework– The two-sample t-test– Checking assumptions, validity
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Homework
• Problem 1.4, pg. 22• Read chapters 1 & 2 and review the remaining
slides in this slideshow.• Be prepared to solve the following in class on
Friday:– 2.4, 2.15, 2.19, 2.25– Helpful hints:
• An Excel data set is available through the textbook website (www.wiley.com/college/montgomery)
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An Example:
• Portland cement formulation – See data, page 24
• ch 2 examples.xls
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Graphical view of the data
• Dot diagram• fig. 2.1, pp. 24
• What do you see?
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If you have a large sample, a histogram may be useful
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• Box plots– Fig. 2.3, pp. 26
• What do you see here?
Another graphical comparison
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The hypothesis testing framework
• Statistical hypothesis testing is a useful framework for many experimental situations
• Origins of the methodology date from the early 1900s
• For the Portland cement example, we will use the two-sample t-test
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The hypothesis testing framework
• Sampling from a normal distribution• Statistical hypotheses:
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0 1 2
1 1 2
::
HH
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Estimation of parameters
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1
2 2 2
1
1 estimates the population mean
1 ( ) estimates the variance 1
n
ii
n
ii
y yn
S y yn
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Portland cement example …
• Summary statistics (see pg. 36)– Can get these from Minitab, as well …
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1
21
1
1
16.76
0.1000.31610
y
SSn
2
22
2
2
17.04
0.0610.24810
y
SSn
Modified Mortar Unmodified Mortar
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The two-sample t-test:
• Use the sample mean to draw inferences about the population means …
• Comparison is …
• This suggests a statistic,
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_____________________21 yy
ny
22 where
,means the in difference the of deviation standard
meansinsdifference
2
22
1
21
210
nn
yyZ
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The two-sample t-test:
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2 2 2 21 2 1 2
1 22 2
1 2
1 2
2 2 21 2
2 22 1 1 2 2
1 2
Use and to estimate and
The previous ratio becomes
However, we have the case where Pool the individual sample variances:
( 1) ( 1)2p
S Sy y
S Sn n
n S n SSn n
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The two-sample t-test:
• Values of t0 that are near zero are consistent with the null hypothesis
• Values of t0 that are very different from zero are consistent with the alternative hypothesis
• t0 is a “distance” measure-how far apart the averages are expressed in standard deviation units
• Note: this is an interpretation of t0 as a signal-to-noise ratioISE 352 - Ch. 1&2 19
1 20
1 2
The test statistic is
1 1
p
y ytS
n n
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For the Portland cement example
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2 22 1 1 2 2
1 2
1 20
1 2
( 1) ( 1) 9(0.100) 9(0.061) 0.0812 10 10 2
0.284
16.76 17.04 2.201 1 1 10.284
10 10
The two sample means are a little over two standard deviations apartIs t
p
p
p
n S n SSn n
S
y ytS
n n
his a "large" difference?
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The two-sample (pooled) t-test• So far, we haven’t really done any “statistics”• We need an objective basis for deciding how large the test
statistic t0 really is– In 1908, W. S. Gosset derived the reference distribution for t0 …
called the t distribution– Tables of the t distribution – see Appendix II, pg. 614– In Excel, use the function
» ________________________________________
• OR …
– In Minitab, …
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The two-sample (pooled) t-test
• The P-value is the area (probability) in the tails of the t-distribution beyond -2.20 + the probability beyond +2.20 (it’s a two-sided test)
• The P-value is a measure of how unusual the value of the test statistic is given that the null hypothesis is true
• The P-value the risk of wrongly rejecting the null hypothesis of equal means (it measures rareness of the event)
• The P-value in our problem is P = 0.042• (Note the values for tcrit used in the traditional approach)
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p = 0.042
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Computer two-sample t-test results
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t-Test: Two-Sample Assuming Equal Variances
Modified UnmodifiedMean 16.764 17.042Variance 0.100138 0.061462222Observations 10 10Pooled Variance 0.0808Hypothesized Mean Difference 0df 18t Stat -2.18688P(T<=t) one-tail 0.021098t Critical one-tail 1.734064P(T<=t) two-tail 0.042197t Critical two-tail 2.100922
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Checking assumptions – the normal probability plot
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Normality testing in Minitab
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Importance of the t-Test
• Provides an objective framework for simple comparative experiments
• Could be used to test all relevant hypotheses in a two-level factorial design, because all of these hypotheses involve the mean response at one “side” of the cube versus the mean response at the opposite “side” of the cube
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Confidence intervals (See pg. 44)
• Hypothesis testing gives an objective statement concerning the difference in means, but it doesn’t specify “how different” they are
• General form of a confidence interval
• The 100(1- α)% confidence interval on the difference in two means:
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where ( ) 1 L U P L U
1 2
1 2
1 2 / 2, 2 1 2 1 2
1 2 / 2, 2 1 2
(1/ ) (1/ )
(1/ ) (1/ )
n n p
n n p
y y t S n n
y y t S n n
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The Portland cement example(see pg. 45)
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Other chapter topics
• Hypothesis testing when the variances are known
• One sample inference• Hypothesis tests on variances• Paired experiments
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Homework (due Monday, Aug. 27)
• Finish and format the results of the problems you solved in class on Friday.
• Read Chapter 3 and review the PowerPoint slides.
• Be prepared to solve the following in class:– 3.3– 3.8, part a only (use Minitab or Excel)
• Bring your solutions to class on Monday.
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