1. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-1 Chapter 7 Confidence Interval Estimation Statistics for Managers Using Microsoft ® Excel 4 th Edition

2. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-2 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 error

3. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-3 Point and Interval Estimates  A point estimate is a single number,  a confidence interval provides additional information about variability Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval

4. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-4 We can estimate a Population Parameter … Point Estimates with a Sample Statistic (a Point Estimate) Mean Proportion psps p X μ

5. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-5 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% confident

6. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-6 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 interval

7. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-7 Estimation Process (mean, μ, is unknown) Population Random Sample Mean X = 50 Sample I am 95% confident that μ is between 40 & 60.

8. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-8 Confidence Intervals Population Mean σ Unknown t Distribution Confidence Intervals Population Proportion Normal Distribution Z σ Known Normal Distribution

9. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-9 Intervals and Level of Confidence Confidence Intervals Sampling Distribution of the Mean x x1x1 x2x2 Confidence

10. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-10  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 distribution Confidence Interval for μ (σ Unknown)

11. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-11 Student’s t Distribution t 0 t (df = 5) t (df = 13) t-distributions are bell- shaped and symmetric, but have ‘fatter’ tails than the normal Standard Normal (t with df > 30) Note: t Normal as n increases

12. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-12  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: (where t is the critical value of the t distribution with n-1 d.f. and an area of α/2 in each tail) Confidence Interval for μ (σ Unknown) (continued)

13. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-13 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 The confidence interval is 46.698 ….. ….. 53.302 46.698   53.302  

14. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-14 Example d.f. = n – 1 = 24, so 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 2.0639

15. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-15 Confidence Intervals Population Mean σ Unknown Confidence Intervals Population Proportion σ Known

16. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-16 Confidence Intervals for the Population Proportion, p  Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation  We will estimate this with sample data: (continued)

17. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-17 Confidence Interval Endpoints  Upper and lower confidence limits for the population proportion are calculated with the formula  To get a Z value use the NORMSINV function with alpha/2 for 95% confidence use.025  Result 1.96

18. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-18 Example  A random sample of 100 people shows that 25 are left-handed.  Form a 95% confidence interval for the true proportion of left-handers

19. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-19 Example  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. 2. 3. (continued)

20. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-20 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.

21. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-21 Determining Sample Size For the Mean Determining Sample Size For the Proportion

22. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-22 Determining Sample Size For the Mean Determining Sample Size Sampling error (margin of error)

23. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-23 Determining Sample Size For the Mean Determining Sample Size (continued) Now solve for n to get

24. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-24 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, s

25. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-25 Determining Sample Size Determining Sample Size For the Proportion Sampling error (margin of error)

26. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-26 Determining Sample Size Determining Sample Size For the Proportion Now solve for n to get (continued)

27. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-27 PHStat Interval Options options

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

29. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-29 Using PHStat (for μ, σ unknown) A random sample of n = 25 has X = 50 and S = 8. Form a 95% confidence interval for μ

30. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-30 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 p s =.12)

31. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-31 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 auditors

32. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-32 Confidence Interval for Population Total Amount  Point estimate:  Confidence interval estimate: (This is sampling without replacement, so use the finite population correction in the confidence interval formula)

33. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-33 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.

34. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-34  Point estimate:  Where the average difference, D, is: Confidence Interval for Total Difference

35. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-35  Confidence interval estimate: where Confidence Interval for Total Difference (continued)

36. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-36 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 provided

37. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-37 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 samples

38. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-38 Chapter Summary  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 issues (continued)