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Statistics for Managers                                      Using Microsoft® Excel                                       ...
Chapter Goals     After completing this chapter, you should be able to:      Distinguish between a point estimate and an ...
Point and Interval Estimates                 A point estimate is a single number,                 a confidence interval ...
Point Estimates                    We can estimate a                                            with a Sample             ...
Confidence Interval Estimate             An interval gives a range of values:                     Takes into considerati...
Confidence Level, (1- )                    Suppose confidence level = 95%                    Also written (1 - ) = .95  ...
Estimation Process                                                     Random Sample             I am 95%                 ...
Confidence Intervals                                                                Confidence                            ...
Intervals and Level of Confidence                                        Sampling Distribution of the Mean                ...
Confidence Interval for μ                                                      (σ Unknown)            If the population s...
Student’s t Distribution                               Note: t                          Normal as n increases             ...
Confidence Interval for μ                                                       (σ Unknown)                               ...
Example                 A random sample of n = 25 has X = 50 and                 S = 8. Form a 95% confidence interval for...
Example                   d.f. = n – 1 = 24, so t                                     /2 , n 1   t.025,24   2.0639        ...
Confidence Intervals                                                                Confidence                            ...
Confidence Intervals for the                          Population Proportion, p                                            ...
Confidence Interval Endpoints          Upper and lower confidence limits for the           population proportion are calc...
Example                   A random sample of 100 people shows                    that 25 are left-handed.                ...
Example                                                                                               (continued)         ...
Interpretation                We are 95% confident that the true                 percentage of left-handers in the popula...
Determining Sample Size                                                               Determining                         ...
Determining Sample Size                                                               Determining                         ...
Determining Sample Size                                                                                        (continued)...
If σ is unknown                If σ is unknown it can be estimated                 from experience or                Sel...
Determining Sample Size                                                               Determining                         ...
Determining Sample Size                                                                                            (contin...
PHStat Interval Options                                                                               optionsStatistics fo...
PHStat Sample Size OptionsStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.   Chap 7-28
Using PHStat                                               (for μ, σ unknown)                 A random sample of n = 25 ha...
Using PHStat                                 (sample size for proportion)                     How large a sample would be ...
Applications in Auditing             Advantages of statistical sampling in auditing                     Sample result is...
Confidence Interval for                                 Population Total Amount             Point estimate:              ...
Confidence Interval for                              Population Total: Example                An firm has a population of ...
Confidence Interval for                                       Total Difference             Point estimate:               ...
Confidence Interval for                                       Total Difference                                            ...
Ethical Issues             A confidence interval (reflecting sampling error)              should always be reported along...
Chapter Summary                 Introduced the concept of confidence intervals                 Discussed point estimates...
Chapter Summary                                                                               (continued)                ...
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Sampling Excel

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Sampling Excel

  1. 1. Statistics for Managers Using Microsoft® Excel 4th Edition Chapter 7 Confidence Interval EstimationStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-1
  2. 2. Chapter Goals After completing this chapter, you should be able to:  Distinguish between a point estimate and an interval estimate  Construct and interpret a confidence interval estimate for a population mean using the t distribution  Form and interpret a confidence interval estimate for a population proportion using the Z distribution  Determine the required sample size to estimate a mean or proportion within a specified margin of errorStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-2
  3. 3. Point and Interval Estimates  A point estimate is a single number,  a confidence interval provides additional information about variability Lower Upper Confidence Confidence Point Estimate Limit Limit Width of confidence intervalStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-3
  4. 4. Point Estimates We can estimate a with a Sample Population Parameter … Statistic (a Point Estimate) Mean μ X Proportion p psStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-4
  5. 5. Confidence Interval Estimate  An interval gives a range of values:  Takes into consideration the variation in sample statistics from sample to sample  Based on observation from 1 sample  Gives information about closeness to unknown population parameters  Stated in terms of level of confidence  Can never be 100% confidentStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-5
  6. 6. Confidence Level, (1- )  Suppose confidence level = 95%  Also written (1 - ) = .95  Where is the risk of being wrong  A relative frequency interpretation:  In the long run, 95% of all the confidence intervals that can be constructed will contain the unknown parameter  A specific interval either will contain or will not contain the true parameter  No probability involved in a specific intervalStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-6
  7. 7. Estimation Process Random Sample I am 95% confident that μ is between Population Mean 40 & 60. (mean, μ, is X = 50 unknown) SampleStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-7
  8. 8. Confidence Intervals Confidence Intervals Population Population Mean Proportion Normal Distribution Z σ Known σ Unknown Normal t Distribution DistributionStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-8
  9. 9. Intervals and Level of Confidence Sampling Distribution of the Mean /2 1 /2 x μx μ Confidence x1 x2 Confidence IntervalsStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-9
  10. 10. Confidence Interval for μ (σ Unknown)  If the population standard deviation σ is unknown, we can substitute the sample standard deviation, s as an estimate  This introduces extra uncertainty, since s is different from sample to sample  In these circumstances the t distribution is used instead of the normal distributionStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-10
  11. 11. Student’s t Distribution Note: t Normal as n increases Standard Normal (t with df > 30) t (df = 13) t-distributions are bell- shaped and symmetric, but have ‘fatter’ tails than the t (df = 5) normal 0 tStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-11
  12. 12. Confidence Interval for μ (σ Unknown) (continued)  Assumptions  Population standard deviation is unknown  Population is not highly skewed  Population is normally distributed or the sample size is large (>30)  Use Student’s t Distribution  Confidence Interval Estimate: S X t n-1 n (where t is the critical value of the t distribution with n-1 d.f. and anStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-12
  13. 13. Example A random sample of n = 25 has X = 50 and S = 8. Form a 95% confidence interval for μ  d.f. = n – 1 = 24, so t /2 , n 1 t.025,24 2.0639 The confidence interval is S 8 X t /2, n-1 50 (2.0639) n 25 46.698 ….. ….. 53.302 46.698 53.302Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-13
  14. 14. Example d.f. = n – 1 = 24, so t /2 , n 1 t.025,24 2.0639 To get a t value use the TINV function. The value of alpha (1-confidence)/2 and n-1 degrees of freedom are the inputs needed. For 95% confidence use .025 and for a sample size of 25 use 24 df Result 2.0639Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-14
  15. 15. Confidence Intervals Confidence Intervals Population Population Mean Proportion σ Known σ UnknownStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-15
  16. 16. Confidence Intervals for the Population Proportion, p (continued)  Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation p(1 p) σp n  We will estimate this with sample data: ps(1 ps ) S ps nStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-16
  17. 17. Confidence Interval Endpoints  Upper and lower confidence limits for the population proportion are calculated with the formula ps(1 ps ) ps Z n  To get a Z value use the NORMSINV function with alpha/2 for 95% confidence use .025  Result 1.96Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-17
  18. 18. Example  A random sample of 100 people shows that 25 are left-handed.  Form a 95% confidence interval for the true proportion of left-handersStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-18
  19. 19. Example (continued)  A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers. 1. ps 25/100 .25 Sps ps(1 ps )/n .25(.75)/1 00 .0433 2. 3. .25 1.96 (.0433) 0.1651 p 0.3349Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-19
  20. 20. Interpretation  We are 95% confident that the true percentage of left-handers in the population is between 16.51% and 33.49%.  Although this range may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-20
  21. 21. Determining Sample Size Determining Sample Size For the For the Mean ProportionStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-21
  22. 22. Determining Sample Size Determining Sample Size For the Mean Sampling error (margin of error) σ σ X Z e Z n nStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-22
  23. 23. Determining Sample Size (continued) Determining Sample Size For the Mean σ Now solve Z σ 2 2 e Z for n to get n 2 n eStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-23
  24. 24. If σ is unknown  If σ is unknown it can be estimated from experience or  Select a pilot sample and estimate σ with the sample standard deviation, sStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-24
  25. 25. Determining Sample Size Determining Sample Size For the Proportion ps(1 ps ) p(1 p ) ps Z e Z n n Sampling error (margin of error)Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-25
  26. 26. Determining Sample Size (continued) Determining Sample Size For the Proportion p(1 p ) Now solve Z 2 p (1 p)e Z for n to get n 2 n eStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-26
  27. 27. PHStat Interval Options optionsStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-27
  28. 28. PHStat Sample Size OptionsStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-28
  29. 29. Using PHStat (for μ, σ unknown) A random sample of n = 25 has X = 50 and S = 8. Form a 95% confidence interval for μStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-29
  30. 30. Using PHStat (sample size for proportion) How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence? (Assume a pilot sample yields ps = .12)Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-30
  31. 31. Applications in Auditing  Advantages of statistical sampling in auditing  Sample result is objective and defensible  Sample size estimation is done in advance on an objective basis  Provides an estimate of the sampling error  Can provide more accurate conclusions than a census of the population  Samples can be combined and evaluated by different auditorsStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-31
  32. 32. Confidence Interval for Population Total Amount  Point estimate: Population total NX  Confidence interval estimate: S N n NX N( t n 1 ) n N 1 (This is sampling without replacement, so use the finite population correction in the confidence interval formula)Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-32
  33. 33. Confidence Interval for Population Total: Example An firm has a population of 1000 accounts and wishes to estimate the total population value. A sample of 80 accounts is selected with average balance of $87.6 and standard deviation of $22.3. Find the 95% confidence interval estimate of the total balance.Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-33
  34. 34. Confidence Interval for Total Difference  Point estimate: Total Difference ND  Where the average difference, D, is: n Di i 1 D n where Di audited value - original valueStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-34
  35. 35. Confidence Interval for Total Difference (continued)  Confidence interval estimate: SD N n ND N( t n 1 ) n N 1 where n (Di D)2 i 1 SD n 1Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-35
  36. 36. Ethical Issues  A confidence interval (reflecting sampling error) should always be reported along with a point estimate  The level of confidence should always be reported  The sample size should be reported  An interpretation of the confidence interval estimate should also be providedStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-36
  37. 37. Chapter Summary  Introduced the concept of confidence intervals  Discussed point estimates  Developed confidence interval estimates  Determined confidence interval estimates for the mean (σ unknown)  Created confidence interval estimates for the proportion  Determined required sample size for mean and proportion estimation samplesStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-37
  38. 38. Chapter Summary (continued)  Developed applications of confidence interval estimation in auditing  Confidence interval estimation for population total  Confidence interval estimation for total difference in the population  Addressed confidence interval estimation and ethical issuesStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-38

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