ENMA 420/520Statistical ProcessesSpring 2007

Michael F. Cochrane, Ph.D.

Dept. of Engineering Management

Old Dominion University

Class EightReadings & Problems

Continuing assignment from last week!

Reading assignment

M & S Chapter 7 Sections 7.1 – 7.7; 7.9, 7.11

Recommended problems

M & S Chapter 7 37, 40 (use Excel), 47, 61, 85, 98, 104

Estimating y2:

Convenience of Normality Now Absent!

Recall that s2 is a scaled X2 distribution

Same approach for estimation

Take sample of n observations

Use s20 as basis for estimating 2

y

point estimate confidence interval

A Normal Sampling Distribution

0.0

0.1

0.2

5 9 13 17 21 25

What are the casesin which the samplingdistribution is “conveniently” normal?

Now want to estimate 2y

s^2~Chisq

0.00

0.05

0.11

0.0 6.5 12.9 19.4 25.9 32.4

Getting Confidence Interval for y2:

Conceptually Same Approach

2

22 )1(

y

sn

Recall from Section 6.11

Which variables are random variables?

Here is conceptual approach to be taken:- sample n observations- calculate s0

2 from sample- substitute for X0

2 in terms of s02 and y

2 in the following

p(X2(1-/2) X0

2 X2(/2) ) = 1 -

- the above range provides the (1- )100% CI for y

2

What is thereasoning behind this?

The (1- )100% CI for y2:

Working Through the Math

))1()1(

(

))1(

(

)(1

2)21(

202

22

20

222

22

)21(

22

22)21(

0

snsnp

snp

p

y

y

Why?

Why?

Where do we get these?

Notes: - the parent distribution y is assumed normal- the CI is not necessarily symmetric about s2

Estimating CI for y2:

Example Problem

Problem summary

Took n = 10 observations

Found s0= 0.0098

Want 95% CI for y2

This is pdf of s2,a scaled X2 distribution

s^2 = y2

This area is 0.05 What are the critical values on the pdf?

Previous Example ProblemFinding the 95% CI

]0003216.00000457.0[

]70039.2

)0098.0)(9(

0228.19

)0098.0)(9([

))1()1(

(1

2

22

2

2)21(

202

22

20

y

y

y

snsnp

How do you interpret the above confidence interval?

Your sample variance was 0.00009604, do you see that theCI is not symmetric about your sample statistic?

Thinking About the Solutionto Example Problem

This is THE pdf of s2,a scaled X2 distribution.For n=10, it exists and isexact.

What keeps you fromdetermining it exactly?

s^2 = y2

This is s^2 which you do not know,but you wish you did.

This area is 0.05,how often will your sample s2

fall in this range?

This is the width of the CI,

the actual CI will depend on your sample.

Estimating Relationship Between Variances of Two Populations

For means estimated differences between population means

Why not estimate difference between population variances?

Do you recall Section 6.11 in text?

2

2y

2

1 estimate Willy

Ratio of VarianceTwo Populations

F distribution has 2 associated degrees of freedom

1 = n1 - 1 ==> associated with numerator

2 = n2 - 1 ==> associated with denominator

Have tabulated values of F (1, 2)

Excel provides significantly more capability than tables

))((

)1(but

21

22

22

21

2

22

2

22

1

21

s

sF

snF

Which are the random variables?

The F distributionis a “standard”distribution

CI for the Ratio of VariancesFrom Two Populations

Let’s discuss above CI and use of Table in text

Problem 7.78 in M&S

]1

[

for CI %100)1(

),(2/22

21

2y

2y

),(2/22

21

2y

2y

12

2

1

21

2

1

Fs

s

Fs

s

Take note of all variables

Illustrating CI ofRatio of Population Variances

Problem 7.79

Comparing shear stress variances for two types of wood

Southern Pine

N = 100, y-bar = 1312, s = 422

Ponderosa Pine

N = 47, y-bar = 1352, s = 271

Use interval estimation to Compare variation in shear stresses Draw inference from analysis

Choosing Sample Size

How many measurements should we include in our sample??

Must ask these questions:

How wide do we want our CI to be?

What confidence coefficient do we require?

Also a function of cost of sampling!

Also a function of cost of sampling!

Choosing Sample Size

Based on CI “half-width”, H

n

stH

2

Small sample half-width for pop. mean

We don’t know “s”, so we’ll have to approximate

See example 7.17 on page 315

Sample size for population proportion

If no estimate of “p” available, use p = q = 0.5If true p value differs substantially from 0.5, you’ll have a larger sample than needed

pqH

zn

2

2

Recall our polling

example… H is the “margin of error”