Business Statistics 3 e by Ken Black Chapter

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

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. Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 2

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

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

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

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

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

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

Distribution of Sample Means for 95% Confidence . 025 95%. 4750 -1. 96 0 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning X 1. 96 Z 9

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

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

95% Confidence Intervals for Is our interval, 95% 3. 98 4. 54, in the red? X X X X Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 12

Demonstration Problem 8. 1 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 13

Demonstration Problem 8. 2 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 14

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

Z Values for Some of the More Common Levels of Confidence Level Z Value 90% 1. 645 95% 1. 96 98% 2. 33 99% 2. 575 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 16

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 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 17

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 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 18

Comparison of Selected t Distributions to the Standard Normal t (d. f. = 25) t (d. f. = 1) -3 -2 -1 0 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 1 2 3 19

Table of Critical Values of t df 1 2 3 4 5 t 0. 100 t 0. 050 t 0. 025 t 0. 010 t 0. 005 3. 078 1. 886 1. 638 1. 533 1. 476 6. 314 2. 920 2. 353 2. 132 2. 015 12. 706 4. 303 3. 182 2. 776 2. 571 31. 821 6. 965 4. 541 3. 747 3. 365 63. 656 9. 925 5. 841 4. 604 4. 032 1. 714 25 1. 319 1. 318 1. 316 1. 708 2. 069 2. 064 2. 060 2. 500 2. 492 2. 485 2. 807 2. 797 2. 787 29 30 1. 311 1. 310 1. 699 1. 697 2. 045 2. 042 2. 462 2. 457 2. 756 2. 750 40 60 120 1. 303 1. 296 1. 289 1. 282 1. 684 1. 671 1. 658 1. 645 2. 021 2. 000 1. 980 1. 960 2. 423 2. 390 2. 358 2. 327 2. 704 2. 660 2. 617 2. 576 23 24 1. 711 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning t 20

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

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

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

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

Solution for Demonstration Problem 8. 5 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 25

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. Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 26

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

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

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

2 Table df 0. 975 0. 950 1 9. 82068 E-04 3. 93219 E-03 2 0. 0506357 0. 102586 3 0. 2157949 0. 351846 4 0. 484419 0. 710724 5 0. 831209 1. 145477 6 1. 237342 1. 63538 7 1. 689864 2. 16735 8 2. 179725 2. 73263 9 2. 700389 3. 32512 10 3. 24696 3. 94030 0. 100 2. 70554 4. 60518 6. 25139 7. 77943 9. 23635 10. 6446 12. 0170 13. 3616 14. 6837 15. 9872 0. 050 3. 84146 5. 99148 7. 81472 9. 48773 11. 07048 12. 5916 14. 0671 15. 5073 16. 9190 18. 3070 0. 025 5. 02390 7. 37778 9. 34840 11. 14326 12. 83249 14. 4494 16. 0128 17. 5345 19. 0228 20. 4832 20 21 22 23 24 25 28. 4120 29. 6151 30. 8133 32. 0069 33. 1962 34. 3816 31. 4104 32. 6706 33. 9245 35. 1725 36. 4150 37. 6525 34. 1696 35. 4789 36. 7807 38. 0756 39. 3641 40. 6465 70 80 90 100 9. 59077 10. 28291 10. 9823 11. 6885 12. 4011 13. 1197 10. 8508 11. 5913 12. 3380 13. 0905 13. 8484 14. 6114 48. 7575 51. 7393 85. 5270 90. 5313 95. 0231 57. 1532 60. 3915 96. 5782 101. 8795 106. 6285 65. 6466 69. 1260 107. 5650 113. 1452 118. 1359 74. 2219 77. 9294 124. 3421 Learning 129. 5613 Business Statistics: Contemporary Decision Making, 3 e, by 118. 4980 Black. © 2001 South-Western/Thomson df = 5 0. 10 0 5 10 15 20 9. 23635 30

Two Table Values of 2 df = 7. 05. 95. 05 0 2 4 6 8 2. 16735 10 12 14 16 18 20 df 1 2 3 4 5 6 7 8 9 10 0. 950 3. 93219 E-03 0. 102586 0. 351846 0. 710724 1. 145477 1. 63538 2. 16735 2. 73263 3. 32512 3. 94030 0. 050 3. 84146 5. 99148 7. 81472 9. 48773 11. 07048 12. 5916 14. 0671 15. 5073 16. 9190 18. 3070 20 21 22 23 24 25 10. 8508 11. 5913 12. 3380 13. 0905 13. 8484 14. 6114 31. 4104 32. 6706 33. 9245 35. 1725 36. 4150 37. 6525 14. 0671 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 31

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

Solution for Demonstration Problem 8. 6 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 33

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

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

Solution for Demonstration Problem 8. 7 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 36

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

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

Determining Sample Size when Estimating P with No Prior Information P PQ 0. 5 0. 25 0. 4 0. 24 0. 3 0. 21 0. 2 0. 16 0. 1 0. 09 Z = 1. 96 E = 0. 05 400 350 300 250 n 200 150 100 50 0 0 0. 1 0. 2 0. 3 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 0. 4 0. 5 P 0. 6 0. 7 0. 8 0. 9 1 39

Solution for Demonstration Problem 8. 9 Business Statistics: Contemporary Decision Making, 3 e, by Black. © 2001 South-Western/Thomson Learning 40