1. ENGR 610 Applied Statistics Fall 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 1 st 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 where Is the standard normal score (“Z-score”) With and effective mean of zero and a standard deviation of 1 68, 95, 99.7%

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

9. Normal Approximation to Poisson Distribution For Poisson distribution and so Variance,, should be at least 5

10. Normal Probability Plot Use normal probability graph paper to plot ordered cumulative percentages, P i = (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 Only memoryless random distribution Poisson, with continuous rate of change,

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

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 (s 2 ) using Student’s t distribution Estimating population variance (  2 ) from sample variance (s 2 ) 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 (s 2 ) 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 known population variance (  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 sample variance (s 2 ) Using Student’s t distribution with n-1 degrees of freedom Where t n-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 (s 2 ) Using  2 distribution with n-1 degrees of freedom Where  U and  L, the upper and lower 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 (s 2 ) 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 (s 2 ) Using k-statistics (Tables A.5a, A.5.b) Where K 1 and K 2 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 to where 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