ENGR 610 Applied Statistics Fall 2007 - Week 4

1. ENGR 610Applied StatisticsFall 2007 - Week 4 Marshall University CITE Jack Smith

2. Overview for Today • Review of Ch 5 • Homework problems for Ch 5 • Estimation Procedures (Ch 8) • Homework assignment • About the 1st exam

3. Chapter 5 Review • Continuous probability distributions • Uniform • Normal • Standard Normal Distribution (Z scores) • Approximation to Binomial, Poisson distributions • Normal probability plot • LogNormal • Exponential • Sampling of the mean, proportion • Central Limit Theorem

4. Continuous Probability Distributions (Mean, expected value) (Variance)

5. Uniform Distribution

6. Normal Distribution Gaussian with peak at µ and inflection points at +/- σ FWHM = 2(2ln(2))1/2 σ

7. Standard Normal Distribution 68,95,99.7% where Is the standard normal score (“Z-score”) With and effective mean of zero and a standard deviation of 1

8. Normal Approximation to Binomial Distribution • For binomial distributionand so • Variance, 2, should be at least 10

9. Normal Approximation to Poisson Distribution • For Poisson distributionand so • Variance, , should be at least 5

10. Normal Probability Plot • Use normal probability graph paperto plot ordered cumulative percentages, Pi = (i - 0.5)/n * 100%, as Z-scores- or - • Use Quantile-Quantile plot (see directions in text)- or - • Use software (PHStat)!

11. Lognormal Distribution

12. Exponential Distribution Poisson, with continuous rate of change,  Only memoryless random distribution

13. Sampling Distribution of the Mean, Proportion • Central Limit Theorem Continuous data (proportion) Attribute data

14. Homework Problems (Ch 5) • 5.66 • 5.67 • 5.68 • 5.69

15. Estimation Procedures • Estimating population mean () • from sample mean (X-bar) and population variance (2) using Standard Normal Z distribution • from sample mean (X-bar) and sample variance (s2) using Student’s t distribution • Estimating population variance (2) • from sample variance (s2) using 2 distribution • Estimating population proportion () • from sample proportion (p) and binomial variance (npq)using Standard Normal Z distribution

16. Estimation Procedures, cont’d • Predicting future individual values (X) • from sample mean (X-bar) and sample variance (s2)using Student’s t distribution • Tolerance Intervals • One- and two-sided • Using k-statistics

17. Parameter Estimation • Statistical inference • Conclusions about population parameters from sample statistics (mean, variation, proportion,…) • Makes use of CLT, various sampling distributions, and degrees of freedom • Interval estimate • With specified level of confidence that population parameter is contained within • When population parameters are known and distribution is Normal,

18. Point Estimator Properties • Unbiased • Average (expectation) value over all possible samples (of size n) equals population parameter • Efficient • Arithmetic mean most stable and precise measure of central tendency • Consistent • Improves with sample size n

19. Estimating population mean () • From sample mean (X-bar) and knownpopulationvariance (2) • Using Standard Normal distribution (and CLT!) • Where Z, the critical value, corresponds to area of (1-)/2 for a confidence level of (1-)100% • For example, from Table A.2, Z = 1.96 corresponds to area = 0.95/2 = 0.475 for 95% confidence interval, where  = 0.05 is the sum of the upper and lower tail portions

20. Estimating population mean () • From sample mean (X-bar) and samplevariance (s2) • Using Student’s t distribution withn-1degrees of freedom • Where tn-1, the critical value, corresponds to area of (1-)/2 for a confidence level of (1-)100% • For example, from Table A.4, t = 2.0639 corresponds to area = 0.95/2 = 0.475 for 95% confidence interval, where /2 = 0.025 is the area of the upper tail portion, and 24 is the number of degrees of freedom for a sample size of 25

21. Estimating population variance (2) • From sample variance (s2) • Using 2 distribution withn-1degrees of freedom • WhereU and L, the upperandlower critical values, corresponds to areas of /2 and 1-/2 for a confidence level of (1-)100% • For example, from Table A.6, U = 39.364 and L = 12.401 correspond to the areas of 0.975 and 0.025 for 95% confidence interval and 24 degrees of freedom

22. Predicting future individual values (X) • From sample mean (X-bar) and sample variation (s2) • Using Student’s t distribution • Prediction interval • Analogous to

23. Tolerance intervals • An interval that includes at least a certain proportion of measurements with a stated confidence based on sample mean (X-bar) and sample variance (s2) • Using k-statistics (Tables A.5a, A.5.b) • WhereK1 and K2 corresponds to a confidence level of (1-)100% for p100% of measurements and a sample size of n Two-sided Lower Bound Upper Bound

24. Estimating population proportion () • From binomial mean (np) and variation (npq) from sample (size n, and proportion p) • Using Standard Normal Z distribution as approximation to binomial distribution • Analogous towhere p = X/n

25. Homework • Ch 8 • Appendix 8.1 • Problems: 8.43-44

26. Exam #1 (Ch 1-5,8) • Take home • Given out (electronically) after in-class review • Open book, notes • No collaboration - honor system • Use Excel w/ PHStat where appropriate, but • Explain, explain, explain! • Due by beginning of class, Sept 27