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Confidence Interval & Estimation Zulkarnain Ishak PSIE Pasca Sarjana Unsri 2007 Topics • • • • • • • Estimation process Point estimates Interval estimates Confidence interval estimation for the mean ( known) Determining sample size Confidence interval estimation for the mean ( unknown) Confidence interval estimation for the proportion Topics (continued) Applications of confidence interval estimation in auditing Confidence interval estimation for population total Confidence interval estimation for total difference in the population Estimation and sample size determination for finite population Confidence interval estimation and ethical issues Estimation Process Population Random Sample Mean, , is unknown Mean X = 50 Sample I am 95% confident that is between 40 & 60. Points Estimation Estimate Population Parameters … Mean Proportion Variance Difference p with Sample Statistics X PS 1 2 2 S 2 X1 X 2 Interval Estimates Provides range of values Take into consideration variation in sample statistics from sample to sample Based on observation from 1 sample Give information about closeness to unknown population parameters Stated in terms of level of confidence Never 100% sure Confidence Interval Estimates Confidence Intervals Mean Known Proportion Unknown Confidence Interval for ( Known) Assumptions Population standard deviation is known Population is normally distributed If population is not normal, use large sample Confidence interval estimate X Z / 2 n X Z / 2 n Elements of Confidence Interval Estimation Level of confidence Precision (range) Confidence in which the interval will contain the unknown population parameter Closeness to the unknown parameter Cost Cost required to obtain a sample of size n Level of Confidence Denoted by 100 1 % A relative frequency interpretation In the long run, 100 1 % of all the confidence intervals that can be constructed will contain the unknown parameter A specific interval will either contain or not contain the parameter No probability involved in a specific interval Interval and Level of Confidence Sampling Distribution of the _ Mean Z / 2 X Intervals extend from /2 X 1 Z / 2 X /2 X X Z X X 100 1 % of intervals constructed contain ; to X Z X 100 % do Confidence Intervals not. Factors Affecting Interval Width (Precision) Data variation Sample size Measured by X Intervals Extend from X - Z x to X + Z x n Level of confidence 100 1 % © 1984-1994 T/Maker Co. Determining Sample Size (Cost) Too Big: Too small: • Requires too much resources • Won’t do the job Determining Sample Size for Mean What sample size is needed to be 90% confident of being correct within ± 5? A pilot study suggested that the standard deviation is 45. 1.645 45 Z n 2 2 Error 5 2 2 2 2 219.2 220 Round Up Determining Sample Size for Mean in PHStat PHStat | sample size | determination for the mean … Example in excel spreadsheet Confidence Interval for ( Unknown) ] Assumptions Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample Use Student’s t Distribution Confidence Interval Estimate X t / 2,n 1 S S X t / 2,n 1 n n Student’s t Distribution Standard Normal Bell-Shaped Symmetric ‘Fatter’ Tails t (df = 13) t (df = 5) Z 0 t Degrees of Freedom (df ) Number of observations that are free to vary after sample mean has been calculated Example degrees of freedom Mean of 3 numbers is 2 X 1 1 (or any number) X 2 2 (or any number) X 3 3 (cannot vary) = n -1 = 3 -1 =2 Student’s t Table Upper Tail Area df .25 .10 .05 Let: n = 3 df = n - 1 = 2 = .10 /2 =.05 1 1.000 3.078 6.314 2 0.817 1.886 2.920 / 2 = .05 3 0.765 1.638 2.353 t Values 0 2.920 t Example A random sample of n 25 has X 50 and S 8. Set up a 95% confidence interval estimate for S S X t / 2,n 1 X t / 2,n 1 n n 8 8 50 2.0639 50 2.0639 25 25 46.69 53.30 Confidence Interval for ( Unknown) in PHStat PHStat | confidence interval | estimate for the mean, sigma unknown Example in excel spreadsheet Confidence Interval Estimate for Proportion Assumptions Two categorical outcomes Population follows binomial distribution Normal approximation can be used if np 5 and n 1 p 5 Confidence interval estimate pS Z / 2 pS 1 pS p pS Z / 2 n pS 1 pS n Example A random sample of 400 Voters showed 32 preferred Candidate A. Set up a 95% confidence interval estimate for p. ps Z / ps 1 ps p ps Z / n ps 1 ps n .08 1 .08 .08 1 .08 .08 1.96 p .08 1.96 400 400 .053 p .107 Confidence Interval Estimate for Proportion in PHStat PHStat | confidence interval | estimate for the proportion … Example in excel spreadsheet Determining Sample Size for Proportion Out of a population of 1,000, we randomly selected 100 of which 30 were defective. What sample size is needed to be within ± 5% with 90% confidence? Z p 1 p 1.645 0.3 0.7 n 2 2 Error 0.05 227.3 228 2 2 Round Up Determining Sample Size for Proportion in PHStat PHStat | sample size | determination for the proportion … Example in excel spreadsheet Applications in Auditing Six advantages of statistical sampling in auditing Sample result is objective and defensible Based on demonstrable statistical principles Provides sample size estimation in advance on an objective basis Provides an estimate of the sampling error Applications in Auditing Can provide more accurate conclusions on the population than the actual investigation of the population Samples can be combined and evaluated by different auditors Examination of the population can be time consuming and subject to more nonsampling error Samples are based on scientific approach Samples can be treated as if they have been done by a single auditor Objective evaluation of the results is possible Based on known sampling error Confidence Interval for Population Total Amount Point estimate NX Confidence interval estimate NX N t / 2,n1 S n N n N 1 Confidence Interval for Population Total: Example An auditor is faced with a population of 1000 vouchers and wishes to estimate the total value of the population of vouchers. A sample of 50 vouchers is selected with average voucher amount of $1076.39, standard deviation of $273.62. Set up the 95% confidence interval estimate of the total amount for the population of vouchers. Example Solution N 1000 n 50 NX N t / 2,n 1 S n X $1076.39 S $273.62 N n N 1 273.62 1000 50 1000 1076.39 1000 2.0096 1000 1 100 1, 076,390 75,830.85 The 95% confidence interval for the population total amount of the vouchers is between 1,000,559.15, and 1,152,220.85 Example Solution in PHStat PHStat | confidence intervals | estimate for the population total Excel spreadsheet for the voucher example Confidence Interval for Total Difference in the Population Point estimate n Where D difference ND D i 1 i is the sample average n Confidence interval estimate SD ND N t / 2,n1 n n Where SD N n N 1 D D i 1 i n 1 2 Estimation for Finite Population Samples are selected without replacement Confidence interval for the mean ( unknown) X t / 2,n1 S n N n N 1 Confidence interval for proportion pS Z / 2 pS 1 pS n N n N 1 Sample Size Determination for Finite Population Samples are selected without replacement When estimating the mean Z / 2 n0 2 e 2 2 When estimating the proportion Z / 2 p 1 p n0 2 e 2 Ethical Issues Confidence interval (reflects sampling error) should always be reported along with the 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 Chapter Summary Illustrated estimation process Discussed point estimates Addressed interval estimates Discussed confidence interval estimation for the mean ( known) Addressed determining sample size Discussed confidence interval estimation for the mean ( unknown) Discussed confidence interval estimation for the proportion Chapter Summary Addressed applications of confidence interval estimation in auditing (continued) Confidence interval estimation for population total Confidence interval estimation for total difference in the population Addressed estimation and sample size determination for finite population Addressed confidence interval estimation and ethical issues