1. Estimation: Confidence Intervals Based in part on Chapter 6 General Business 704

2. Objectives: Estimation n Distinguish point & interval estimates n Explain interval estimates n Compute confidence interval estimates l Population mean & proportion l Population total & difference n Determine necessary sample size

3. Thinking Challenge Suppose you’re interested in the average amount of money that students in this class (the population) have in their possession. How would you find out?

4. Statistical Methods

5. Estimation Process Mean, , is unknown Population Random Sample I am 95% confident that  is between 40 & 60. Mean  X = 50 Sample

6. Population Parameter Estimates

7. Estimation Methods

9. Point Estimation n Provides single value l Based on observations from 1 sample n Gives no information about how close value is to the unknown population parameter Example: Sample mean  X = 3 is point estimate of unknown population mean Example: Sample mean  X = 3 is point estimate of unknown population mean

10. Estimation Methods

11. Interval Estimation n Provides range of values l Based on observations from 1 sample n Gives information about closeness to unknown population parameter l Stated in terms of probability n Example: Unknown population mean lies between 40 & 60 with 95% confidence

12. Key Elements of Interval Estimation Confidence interval Sample statistic (point estimate) Confidence limit (lower) Confidence limit (upper) A probability that the population parameter falls somewhere within the interval.

13. Confidence Limits for Population Mean Parameter = Statistic ± Error © 1984-1994 T/Maker Co.

14. Many Samples Have Same Interval 90% Samples 95% Samples 99% Samples  +1.65   x  +2.58   x  x_ XXXX  +1.96   x  -2.58   x  -1.65   x  -1.96   x   X  =  ± Z   x

15. n Probability that the unknown population parameter falls within interval Denoted (1 -  Denoted (1 -   is probability that parameter is not within interval  is probability that parameter is not within interval n Typical values are 99%, 95%, 90% Level of Confidence

16. Intervals & Level of Confidence Sampling Distribution of Mean Large number of intervals Intervals extend from  X - Z   X to  X + Z   X (1 -  ) % of intervals contain .  % do not.

17. Factors Affecting Interval Width n Data dispersion Measured by  Measured by  n Sample size   X =  /  n   X =  /  n Level of confidence (1 -  ) Level of confidence (1 -  ) l Affects Z Intervals extend from  X - Z   X to  X + Z   X © 1984-1994 T/Maker Co.

18. Confidence Interval Estimates

20. Confidence Interval Mean (  Known) n Assumptions l Population standard deviation is known l Population is normally distributed If not normal, can be approximated by normal distribution (n  30) If not normal, can be approximated by normal distribution (n  30) n Confidence interval estimate Note: 99% Z=2.58, 95% Z=1.96, 90% Z=1.65

21. Estimation Example Mean (  Known) The mean of a random sample of n = 25 is  X = 50. Set up a 95% confidence interval estimate for  if  = 10.

22. Thinking Challenge You’re a Q/C inspector for Gallo. The  for 2-liter bottles is.05 liters. A random sample of 100 bottles showed  X = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles? 2 liter © 1984-1994 T/Maker Co.

23. Confidence Interval Solution for Gallo

24. Confidence Interval Estimates

25. Confidence Interval Mean (  Unknown) n Assumptions l Population standard deviation is unknown l Population must be normally distributed n Use Student’s t distribution n Confidence interval estimate

26. Student’s t Distribution 0 t (df = 5) Standard normal t (df = 13) Bell- shaped Symmetric ‘Fatter’ tails Note: As d.f. approach 120, Z and t become very similar

27. Student’s t Table Assume: n = 3 df= n - 1 = 2  =.10  /2 =.05 2.920 t values  / 2.05

28. Degrees of Freedom n Number of observations that are free to vary after sample statistic has been calculated n Example l Sum of 3 numbers is 6 X 1 = 1 (or any number) X 2 = 2 (or any number) X 3 = 3 (cannot vary) Sum = 6 degrees of freedom = n -1 = 3 -1 = 2

29. Estimation Example Mean (  Unknown) A random sample of n = 25 has  X = 50 & S = 8. Set up a 95% confidence interval estimate for .

30. Thinking Challenge You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time?

31. Confidence Interval Solution for Time Study  X = 3.7 S = 3.8987 S = 3.8987 n = 6, df = n - 1 = 6 - 1 = 5 n = 6, df = n - 1 = 6 - 1 = 5 S /  n = 3.8987 /  6 = 1.592 S /  n = 3.8987 /  6 = 1.592 t.05,5 = 2.0150 t.05,5 = 2.0150 3.7 - (2.015)(1.592)  3.7 + (2.015)(1.592) 3.7 - (2.015)(1.592)  3.7 + (2.015)(1.592) 0.492  6.908 0.492  6.908

32. Confidence Interval Estimates

33. Estimation for Finite Populations n Assumptions l Sample is large relative to population s n / N >.05 n Use finite population correction factor Confidence interval (mean,  unknown) Confidence interval (mean,  unknown)

34. Confidence Interval Estimates

35. Confidence Interval Proportion n Assumptions l Two categorical outcomes l Population follows binomial distribution l Normal approximation can be used  n·p  5 & n·(1 - p)  5 n Confidence interval estimate

36. Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p.

37. Thinking Challenge You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?

38. Confidence Interval Solution for Defects n·p  5 n·(1 - p)  5

39. Estimation Methods

40. Bootstrapping Method n Used if population is not normal n Requires significant computer power n Steps l Take initial sample l Sample repeatedly from initial sample l Compute sample statistic l Form resampling distribution Limits are values that cut off smallest & largest  /2 % Limits are values that cut off smallest & largest  /2 %

41. Finding Sample Sizes For Estimating  I don’t want to sample too much or too little!

42. Sample Size Example What sample size is needed to be 90% confident of being correct within  5? A pilot study suggested that the standard deviation is 45.

43. Thinking Challenge You work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within ± $50. A pilot study showed that  was about $400. What sample size do you use?

44. Sample Size Solution Medical Expenses

45. Finding Sample Sizes For Estimating Proportions I don’t want to sample too much or too little! Remember Error is acceptable error Z is based on confidence level chosen p is the true proportion of “success” Never under-estimate p When in doubt, use p=.5 Remember Error is acceptable error Z is based on confidence level chosen p is the true proportion of “success” Never under-estimate p When in doubt, use p=.5

46. Sample Size Example for Estimating p What sample size is needed to be 90% confident (Z=1.645) of being correct within proportion of.04 when using p=.5 (since no useful estimate of p is available)?

47. Estimation of Population Total n In auditing, population total is more important than mean Total = N  X Total = N  X n Confidence interval (population total) l Degrees of freedom = n - 1

48. Estimation of Differences n Used to estimate the magnitude of errors n Steps l Determine sample size Compute average difference,  D Compute average difference,  D l Compute standard deviation of differences l Set up confidence interval estimate

49. Estimation of Differences Equations Mean Difference: Standard Deviation: Interval Estimate:

50. Objectives: Estimation n Distinguish point & interval estimates n Explain interval estimates n Compute confidence interval estimates l Population mean & proportion l Population total & difference n Determine necessary sample size