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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

- 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

What is the

reasoning

behind this?

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

Why?

Why?

Where do we get these?

Notes: - the parent distribution y is assumed normal

- the CI is not necessarily symmetric about s2

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?

- Problem summary
- Took n = 10 observations
- Found s0= 0.0098

- Want 95% CI for y2

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?

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?

What keeps you from

determining it exactly?

- For means estimated differences between population means
- Why not estimate difference between population variances?

- Do you recall Section 6.11 in text?

The F distribution

is a “standard”

distribution

Which are the

random variables?

- 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

Take note of

all variables

- Let’s discuss above CI and use of Table in text
- Problem 7.78 in M&S

- 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

- 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

- Based on CI “half-width”, H

- We don’t know “s”, so we’ll have to approximate
- See example 7.17 on page 315

- 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”