1. ENGR 610 Applied Statistics Fall 2007 - Week 5 Marshall University CITE Jack Smith

2. Overview for Today Review of Estimation Procedures (Ch 8) Homework problems 8.43-44 Review for Exam #1 Descriptive Statistics (Ch 1-3) Probability Distributions (Ch 4-5)

3. 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

4. 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

5. 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,

6. 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

7. 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

8. 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

9. 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

10. Predicting future individual values (X) From sample mean (X-bar) and sample variation (s 2 ) Using Student’s t distribution Prediction interval Analogous to

11. 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

12. 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

13. Exam #1 (Ch 1-5,8) Take home Open book, notes No collaboration - honor system Two-part Part 1 – no software needed Part 2 – use of Excel or PHStat But, explain, explain, explain! Due at the beginning of class, Sept 27

14. Home problems (Ch 8) 8.43 8.44

15. Review of Ch 1-5 (See prior slides for weeks 1-4)