Chapter 2Based on Design & Analysis of Experiments 7E 2009 Montgomery 1 Design and Analysis of...

Chapter 2 Based on Design & Analysis of Experiments 7E 2009 Montgomery

1

Design and Analysis of Engineering Experiments

Ali Ahmad, PhD

Chapter 2 Design & Analysis of Experiments 7E 2009 Montgomery

2

Chapter 2 –Basic Statistical Methods

• 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


3

Portland Cement Formulation (page 24)


4

Graphical View of the DataDot Diagram, Fig. 2.1, pp. 24


5

If you have a large sample, a histogram may be useful


6

Box Plots, Fig. 2.3, pp. 26


7

The Hypothesis Testing Framework

• Statistical hypothesis testing is a useful framework for many experimental situations

• Origins of the methodology date from the early 1900s

• We will use a procedure known as the two-sample t-test


8

The Hypothesis Testing Framework

• Sampling from a normal distribution• Statistical hypotheses:

0 1 2

1 1 2

:

:

H

H


9

Estimation of Parameters

1

2 2 2

1

1 estimates the population mean

1( ) estimates the variance

1

n

ii

n

ii

y yn

S y yn


10

Summary Statistics (pg. 36)

1

21

1

1

16.76

0.100

0.316

10

y

S

S

n

Formulation 1

“New recipe”

Formulation 2

“Original recipe”

2

22

2

2

17.04

0.061

0.248

10

y

S

S

n


11

How the Two-Sample t-Test Works:

1 2

22y

Use the sample means to draw inferences about the population means

16.76 17.04 0.28

Difference in sample means

Standard deviation of the difference in sample means

This suggests a statistic:

y y

n

1 20 2 2

1 2

1 2

Zy y

n n


12

How the Two-Sample t-Test Works:2 2 2 2

1 2 1 2

1 2

2 21 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 S

y y

S Sn n

n S n SS

n n


13

How the Two-Sample t-Test Works:

• 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

• Notice the interpretation of t0 as a signal-to-noise ratio

1 20

1 2

The test statistic is

1 1

p

y yt

Sn n


14

The Two-Sample (Pooled) t-Test2 2

2 1 1 2 2

1 2

1 20

1 2

( 1) ( 1) 9(0.100) 9(0.061)0.081

2 10 10 2

0.284

16.76 17.04 2.20

1 1 1 10.284

10 10

The two sample means are a little over two standard deviations apart

Is t

p

p

p

n S n SS

n n

S

y yt

Sn n

his a "large" difference?


15

William Sealy Gosset (1876, 1937)

Gosset's interest in barley cultivation led him to speculate that design of experiments should aim, not only at improving the average yield, but also at breeding varieties whose yield was insensitive (robust) to variation in soil and climate.

Developed the t-test (1908)

Gosset was a friend of both Karl Pearson and R.A. Fisher, an achievement, for each had a monumental ego and a loathing for the other.

Gosset was a modest man who cut short an admirer with the comment that “Fisher would have discovered it all anyway.”

http://en.wikipedia.org/wiki/Image:William_Sealy_Gosset.jpg


16

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 textbook appendix

t0 = -2.20


17

The Two-Sample (Pooled) t-Test• A value of t0 between

–2.101 and 2.101 is consistent with equality of means

• It is possible for the means to be equal and t0 to exceed either 2.101 or –2.101, but it would be a “rare event” … leads to the conclusion that the means are different

• Could also use the P-value approach

t0 = -2.20


18

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

t0 = -2.20


19

Computer Two-Sample t-Test Results


20

Checking Assumptions – The Normal Probability Plot


21

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


22

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:

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


23


24

Other Chapter Topics

• Hypothesis testing when the variances are known

• One sample inference• Hypothesis tests on variances• Paired experiments

Date post:	19-Jan-2016
Category:	Documents
Upload:	harry-daniels
View:	221 times
Download:	1 times

Chapter 2Based on Design & Analysis of Experiments 7E 2009 Montgomery 1 Design and Analysis of...

Documents