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ENMA 420/520 Statistical Processes Spring 2007PowerPoint Presentation

ENMA 420/520 Statistical Processes Spring 2007

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### 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

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

- M & S Chapter 7

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

reasoning

behind this?

Getting Confidence Interval for y2:Conceptually Same ApproachRecall 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

Why?

Where do we get these?

The (1- )100% CI for y2: Working Through the MathNotes: - the parent distribution y is assumed normal

- the CI is not necessarily symmetric about 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 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?

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 ProblemWhat keeps you from

determining it exactly?

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?

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

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 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

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

- Comparing shear stress variances for two types of wood

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!

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

- 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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