Enma 420 520 statistical processes spring 2007
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ENMA 420/520 Statistical Processes Spring 2007. Michael F. Cochrane, Ph.D. Dept. of Engineering Management Old Dominion University. Class Eight Readings & Problems. Continuing assignment from last week! Reading assignment M & S Chapter 7 Sections 7.1 – 7.7; 7.9, 7.11 Recommended problems

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ENMA 420/520 Statistical Processes Spring 2007

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Enma 420 520 statistical processes spring 2007

ENMA 420/520Statistical ProcessesSpring 2007

Michael F. Cochrane, Ph.D.

Dept. of Engineering Management

Old Dominion University


Enma 420 520 statistical processes spring 2007

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 y 2 convenience of normality now absent

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 2y

      • point estimate

      • confidence interval

What are the cases

in which the sampling

distribution is

“conveniently” normal?

Now want to estimate 2y


Getting confidence interval for y 2 conceptually same approach

What is the

reasoning

behind this?

Getting Confidence Interval for y2:Conceptually Same Approach

Recall from Section 6.11

Which variables are random variables?

Here is conceptual approach to be taken:

- sample n observations

- calculate s02 from sample

- substitute for X02 in terms of s02 and y2 in the following

p(X2(1-/2)  X02  X2(/2) ) = 1 - 

- the above range provides the (1-)100% CI for y2


The 1 100 ci for y 2 working through the math

Why?

Why?

Where do we get these?

The (1- )100% CI for y2: Working Through the Math

Notes: - the parent distribution y is assumed normal

- the CI is not necessarily symmetric about s2


Estimating ci for y 2 example problem

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?

Estimating CI for y2:Example Problem

  • Problem summary

    • Took n = 10 observations

    • Found s0= 0.0098

  • Want 95% CI for y2


Previous example problem finding the 95 ci

Previous Example ProblemFinding the 95% CI

How do you interpret the above confidence interval?

Your sample variance was 0.00009604, do you see that the

CI is not symmetric about your sample statistic?


Thinking about the solution to example problem

This is the width of the CI,

the actual CI will depend on your sample.

This is THE pdf of s2,

a scaled X2 distribution.

For n=10, it exists and is

exact.

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?

Thinking About the Solutionto Example Problem

What keeps you from

determining it exactly?


Estimating relationship between variances of two populations

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?


Ratio of variance two populations

The F distribution

is a “standard”

distribution

Which are the

random variables?

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


Ci for the ratio of variances from two populations

Take note of

all variables

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


Illustrating ci of ratio of population variances

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

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!


Choosing sample size1

Small sample half-width for pop. mean

Choosing Sample Size

  • Based on CI “half-width”, H

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

  • See example 7.17 on page 315


Sample size for population proportion

Sample size for population proportion

  • If no estimate of “p” available, use p = q = 0.5

  • If true p value differs substantially from 0.5, you’ll have a larger sample than needed

Recall our polling example… H is the “margin of error”


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