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- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. PHStat Interval Options optionsStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-27
- 28. PHStat Sample Size OptionsStatistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-28
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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