1. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-1 Business Statistics, 3e by Ken Black Chapter 8 Statistical Inference: Estimation for Single Populations

2. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-2 Learning Objectives Know the difference between point and interval estimation. Estimate a population mean from a sample mean for large sample sizes. Estimate a population mean from a sample mean for small sample sizes. Estimate a population proportion from a sample proportion. Estimate the population variance from a sample variance. Estimate the minimum sample size necessary to achieve given statistical goals.

3. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-3 Statistical Estimation Point estimate -- the single value of a statistic calculated from a sample Interval Estimate -- a range of values calculated from a sample statistic(s) and standardized statistics, such as the Z. –Selection of the standardized statistic is determined by the sampling distribution. –Selection of critical values of the standardized statistic is determined by the desired level of confidence.

4. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-4 Confidence Interval to Estimate  when n is Large Point estimate Interval Estimate

5. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-5 Distribution of Sample Means for (1-  )% Confidence  X  Z 0

6. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-6 Distribution of Sample Means for (1-  )% Confidence  X Z 0

7. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-7 Distribution of Sample Means for (1-  )% Confidence  X Z 0

8. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-8 Probability Interpretation of the Level of Confidence

9. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-9 Distribution of Sample Means for 95% Confidence .4750 X 95%.025 Z 1.96-1.960

10. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-10 95% Confidence Interval for 

11. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-11 95% Confidence Intervals for   X 95% X X X X X X

12. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-12 95% Confidence Intervals for   X 95% X X X X X X Is our interval, 3.98  4.54, in the red?

13. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-13 Demonstration Problem 8.1

14. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-14 Demonstration Problem 8.2

15. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-15 Confidence Interval to Estimate  when n is Large and  is Unknown

16. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-16 Z Values for Some of the More Common Levels of Confidence 90% 95% 98% 99% Confidence Level Z Value 1.645 1.96 2.33 2.575

17. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-17 Estimating the Mean of a Normal Population: Small n and Unknown  The population has a normal distribution. The value of the population standard deviation is unknown. The sample size is small, n < 30. Z distribution is not appropriate for these conditions t distribution is appropriate

18. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-18 The t Distribution Developed by British statistician, William Gosset A family of distributions -- a unique distribution for each value of its parameter, degrees of freedom (d.f.) Symmetric, Unimodal, Mean = 0, Flatter than a Z t formula

19. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-19 Comparison of Selected t Distributions to the Standard Normal -3-20123 Standard Normal t (d.f. = 25) t (d.f. = 5) t (d.f. = 1)

20. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-20 Table of Critical Values of t df t 0.100 t 0.050 t 0.025 t 0.010 t 0.005 13.0786.31412.70631.82163.656 21.8862.9204.3036.9659.925 31.6382.3533.1824.5415.841 41.5332.1322.7763.7474.604 51.4762.0152.5713.3654.032 231.3191.7142.0692.5002.807 24 1.318 1.711 2.0642.4922.797 251.3161.7082.0602.4852.787 291.3111.6992.0452.4622.756 301.3101.6972.0422.4572.750 401.3031.6842.0212.4232.704 601.2961.6712.0002.3902.660 1201.2891.6581.9802.3582.617 1.2821.6451.9602.3272.576  tt  

21. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-21 Confidence Intervals for  of a Normal Population: Small n and Unknown 

22. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-22 Solution for Demonstration Problem 8.3

23. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-23 Solution for Demonstration Problem 8.3

24. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-24 Confidence Interval to Estimate the Population Proportion

25. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-25 Solution for Demonstration Problem 8.5

26. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-26 Population Variance Variance is an inverse measure of the group’s homogeneity. Variance is an important indicator of total quality in standardized products and services. Managers improve processes to reduce variance. Variance is a measure of financial risk. Variance of rates of return help managers assess financial and capital investment alternatives. Variability is a reality in global markets. Productivity, wages, and costs of living vary between regions and nations.

27. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-27 Estimating the Population Variance Population Parameter   Estimator of     formula for Single Variance

28. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-28 Confidence Interval for  2

29. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-29 Selected  2 Distributions df = 3 df = 5 df = 10 0

30. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-30  2 Table df0.9750.950 0.1000.0500.025 19.82068E-043.93219E-03 2.705543.841465.02390 20.05063570.102586 4.605185.991487.37778 30.21579490.351846 6.251397.814729.34840 40.4844190.710724 7.779439.4877311.14326 50.8312091.145477 9.2363511.0704812.83249 61.2373421.63538 10.644612.591614.4494 71.6898642.16735 12.017014.067116.0128 82.1797252.73263 13.361615.507317.5345 92.7003893.32512 14.683716.919019.0228 103.246963.94030 15.987218.307020.4832 209.5907710.8508 28.412031.410434.1696 2110.2829111.5913 29.615132.670635.4789 2210.982312.3380 30.813333.924536.7807 2311.688513.0905 32.006935.172538.0756 2412.401113.8484 33.196236.415039.3641 2513.119714.6114 34.381637.652540.6465 7048.757551.7393 85.527090.531395.0231 8057.153260.3915 96.5782101.8795106.6285 9065.646669.1260 107.5650113.1452118.1359 10074.221977.9294 118.4980124.3421129.5613 05101520 0.10 df = 5 9.23635

31. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-31 Two Table Values of  2 02468101214161820 df = 7.05.95 2.1673514.0671 df0.950 0.050 13.93219E-03 3.84146 20.102586 5.99148 30.351846 7.81472 40.710724 9.48773 51.145477 11.07048 61.63538 12.5916 72.16735 14.0671 82.73263 15.5073 93.32512 16.9190 103.94030 18.3070 2010.8508 31.4104 2111.5913 32.6706 2212.3380 33.9245 2313.0905 35.1725 2413.8484 36.4150 2514.6114 37.6525

32. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-32 90% Confidence Interval for  2

33. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-33 Solution for Demonstration Problem 8.6

34. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-34 Determining Sample Size when Estimating  Z formula Error of Estimation (tolerable error) Estimated Sample Size Estimated 

35. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-35 Sample Size When Estimating  Example

36. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-36 Solution for Demonstration Problem 8.7

37. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-37 Determining Sample Size when Estimating P Z formula Error of Estimation (tolerable error) Estimated Sample Size

38. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-38 Solution for Demonstration Problem 8.8

39. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-39 Determining Sample Size when Estimating P with No Prior Information P n 0 50 100 150 200 250 300 350 400 00.10.20.30.40.50.60.70.80.91 Z = 1.96 E = 0.05 P 0.5 0.4 0.3 0.2 0.1 PQ 0.25 0.24 0.21 0.16 0.09

40. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-40 Solution for Demonstration Problem 8.9