ENMA 420/520 Statistical Processes Spring 2007

1. ENMA 420/520Statistical ProcessesSpring 2007 Michael F. Cochrane, Ph.D. Dept. of Engineering Management Old Dominion University

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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?

8. 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?

9. 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?

10. 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

11. 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

12. 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

13. 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!

14. 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

15. 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”