Business Statistics 4 e by Ken Black Chapter

Learning Objectives • Know the difference between point and interval estimation. • Estimate a population mean from a sample mean when is known. • Estimate a population mean from a sample mean when is unknown. • 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, 4 e, by Ken Black. © 2003 John Wiley & Sons. 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, 4 e, by Ken Black. © 2003 John Wiley & Sons. 3

Confidence Interval to Estimate when is Known • Point estimate • Interval Estimate Business

Distribution of Sample Means for (1 - )% Confidence 0 Business Statistics, 4 e,

Distribution of Sample Means for 95% Confidence . 025 95%. 4750 -1. 96 Business

95% Confidence Interval for Business Statistics, 4 e, by Ken Black. © 2003 John

95% Confidence Intervals for 95% X X X X Business Statistics, 4 e, by

95% Confidence Intervals for Is our interval, 95% X 143. 22 162. 78, in

Demonstration Problem 8. 1 Business Statistics, 4 e, by Ken Black. © 2003 John

Demonstration Problem 8. 2 Business Statistics, 4 e, by Ken Black. © 2003 John

Confidence Interval to Estimate when n is Large and is Known Business Statistics, 4

Car Rental Firm Example Business Statistics, 4 e, by Ken Black. © 2003 John

Z Values for Some of the More Common Levels of Confidence Level z Value

Estimating the Mean of a Normal Population: Unknown • The population has a normal distribution. • The value of the population standard deviation is unknown. • z distribution is not appropriate for these conditions • t distribution is appropriate Business Statistics, 4 e, by Ken Black. © 2003 John Wiley & Sons. 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, 4 e, by Ken Black. © 2003 John Wiley & Sons. 18

Comparison of Selected t Distributions to the Standard Normal t (d. f. = 25)

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, 4 e, by Ken Black. © 2003 John Wiley & Sons. t With df = 24 and a = 0. 05, ta = 1. 711. 20

Confidence Intervals for of a Normal Population: Unknown Business Statistics, 4 e, by Ken

Solution for Demonstration Problem 8. 3 Business Statistics, 4 e, by Ken Black. ©

Comp Time: Excel Normal View Business Statistics, 4 e, by Ken Black. © 2003

Comp Time: Excel Formula View A B C D E F 1 Comp Time Data 2 6 21 17 20 7 0 3 8 16 29 3 8 12 4 11 9 21 25 15 16 =B 7+B 13*B 9 5 6 n= 7 Mean = 8 S= 9 Std Error = =COUNT(A 2: F 4) =AVERAGE(A 2: F 4) =STDEV(A 2: F 4) =B 8/SQRT(B 6) 10 11 = 0. 1 12 df = =B 6 -1 13 t= =TINV(B 11, B 12) 14 15 =B 7 -B 13*B 9 Business Statistics, 4 e, by Ken Black. © 2003 John Wiley & Sons. 25

Confidence Interval to Estimate the Population Proportion Business Statistics, 4 e, by Ken Black.

Solution for Demonstration Problem 8. 5 Business Statistics, 4 e, by Ken Black. ©

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, 4 e, by Ken Black. © 2003 John Wiley & Sons. 28

Estimating the Population Variance • Population Parameter • Estimator of • formula for Single

Confidence Interval for 2 Business Statistics, 4 e, by Ken Black. © 2003 John

Selected 2 Distributions df = 3 df = 5 df = 10 0 Business

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 9. 59077 10. 28291 10. 9823 11. 6885 12. 4011 13. 1197 10. 8508 11. 5913 12. 3380 13. 0905 13. 8484 14. 6114 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 48. 7575 57. 1532 65. 6466 74. 2219 51. 7393 60. 3915 69. 1260 77. 9294 85. 5270 96. 5782 107. 5650 118. 4980 90. 5313 101. 8795 113. 1452 124. 3421 95. 0231 106. 6285 118. 1359 129. 5613 Business Statistics, 4 e, by Ken Black. © 2003 John Wiley & Sons. df = 5 0. 10 0 5 10 15 20 9. 23635 With df = 5 and a = 0. 10, c 2 = 9. 23635 32

Two Table Values of 2 df = 7. 05. 95. 05 0 2 4 6 8 10 12 14 2. 16735 Business Statistics, 4 e, by Ken Black. © 2003 John Wiley & Sons. 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 33

90% Confidence Interval for 2 Business Statistics, 4 e, by Ken Black. © 2003

Solution for Demonstration Problem 8. 6 Business Statistics, 4 e, by Ken Black. ©

Determining Sample Size when Estimating • z formula • Error of Estimation (tolerable error)

Sample Size When Estimating : Example Business Statistics, 4 e, by Ken Black. ©

Solution for Demonstration Problem 8. 7 Business Statistics, 4 e, by Ken Black. ©

Determining Sample Size when Estimating p • z formula • Error of Estimation (tolerable

Solution for Demonstration Problem 8. 8 Business Statistics, 4 e, by Ken Black. ©

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 Business Statistics, 4 e, by Ken Black. © 2003 John Wiley & Sons. 0. 1 0. 2 0. 3 0. 4 0. 5 P 0. 6 0. 7 0. 8 0. 9 1 41