Confidence Interval Estimation Zulkarnain Ishak PSIE Pasca Sarjana

Confidence Interval & Estimation Zulkarnain Ishak PSIE Pasca Sarjana Unsri 2007 Chap 6 -

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 2

Topics (continued) n Applications of confidence interval estimation in auditing n n 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 3

Estimation Process Population Random Sample Mean, , is unknown Mean X = 50 I am 95% confident that is between 40 & 60. Sample 4

Points Estimation Estimate Population Parameters … with Sample Statistics Mean Proportion Variance Difference 5

Interval Estimates n Provides range of values n n 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 n Never 100% sure 6

Confidence Interval Estimates Confidence Intervals Mean Known Proportion Unknown 7

Confidence Interval for ( Known) n Assumptions n n Population standard deviation is known Population is normally distributed If population is not normal, use large sample Confidence interval estimate 8

Elements of Confidence Interval Estimation n Level of confidence n n Precision (range) n n Confidence in which the interval will contain the unknown population parameter Closeness to the unknown parameter Cost n Cost required to obtain a sample of size n 9

Level of Confidence n n Denoted by A relative frequency interpretation n n In the long run, 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 n No probability involved in a specific interval 10

Interval and Level of Confidence Sampling Distribution of the _ Mean Intervals extend from of intervals constructed contain ; to Confidence Intervals not. do 11

Factors Affecting Interval Width (Precision) n Data variation n n Measured by Sample size Intervals Extend from X - Z x to X + Z x n n Level of confidence n © 1984 -1994 T/Maker Co. 12

Determining Sample Size (Cost) Too Big: Too small: • Requires too much resources • Won’t do the job 13

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. Round Up 14

Determining Sample Size for Mean in PHStat n n PHStat | sample size | determination for the mean … Example in excel spreadsheet 15

Confidence Interval for ( Unknown) ] n Assumptions n n n 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 n 16

Student’s t Distribution Standard Normal Bell-Shaped Symmetric ‘Fatter’ Tails t (df = 13) t (df = 5) Z 0 t 17

Degrees of Freedom (df ) n n Number of observations that are free to vary after sample mean has been calculated Example degrees of freedom n Mean of 3 numbers is 2 = n -1 = 3 -1 =2 18

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 19

Example 20

Confidence Interval for ( Unknown) in PHStat n n PHStat | confidence interval | estimate for the mean, sigma unknown Example in excel spreadsheet 21

Confidence Interval Estimate for Proportion n Assumptions n n Two categorical outcomes Population follows binomial distribution Normal approximation can be used if and Confidence interval estimate n 22

Example A random sample of 400 Voters showed 32 preferred Candidate A. Set up a 95% confidence interval estimate for p. 23

Confidence Interval Estimate for Proportion in PHStat n n PHStat | confidence interval | estimate for the proportion … Example in excel spreadsheet 24

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? Round Up 25

Determining Sample Size for Proportion in PHStat n n PHStat | sample size | determination for the proportion … Example in excel spreadsheet 26

Applications in Auditing n Six advantages of statistical sampling in auditing n Sample result is objective and defensible n n n Based on demonstrable statistical principles Provides sample size estimation in advance on an objective basis Provides an estimate of the sampling error 27

Applications in Auditing n Can provide more accurate conclusions on the population than the actual investigation of the population n n Samples can be combined and evaluated by different auditors n n n 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 n Based on known sampling error 28

Confidence Interval for Population Total Amount n Point estimate n n Confidence interval estimate n 29

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. 30

Example Solution The 95% confidence interval for the population total amount of the vouchers is between 1, 000, 559. 15, and 1, 152, 220. 85 31

Example Solution in PHStat n n PHStat | confidence intervals | estimate for the population total Excel spreadsheet for the voucher example 32

Confidence Interval for Total Difference in the Population n Point estimate n n Where difference is the sample average Confidence interval estimate n n Where 33

Estimation for Finite Population n Samples are selected without replacement n Confidence interval for the mean ( unknown) n n Confidence interval for proportion n 34

Sample Size Determination for Finite Population n Samples are selected without replacement n When estimating the mean n n When estimating the proportion n 35

Ethical Issues n n 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 36

Chapter Summary n n n n 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 37

Chapter Summary n Addressed applications of confidence interval estimation in auditing n n (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 38