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BA 201 Lecture 13 Sample Size Determination and Ethical Issues Topics Confidence Interval Estimation for the Mean ( Unknown) Confidence Interval Estimation for the Proportion Determining Sample Size Confidence Interval Estimation and Ethical Issues Confidence Interval for ( Unknown) Assumptions Population standard deviation is unknown If population is normal, small sample can be used 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) 0 Z t 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 Elements of Confidence Interval Estimation Level of Confidence 1 100% Precision (Sampling Error) Confidence in which the interval will contain the unknown population parameter Closeness to the unknown parameter Cost (Sample Size) 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 Factors Affecting Interval Width (Precision) Data Variation Measured by Sample Size X Intervals Extend from X Z / 2 X to X Z / 2 X Sampling Error = Z / 2 n Level of Confidence 100 1 % © 1984-1994 T/Maker Co. X Determining Sample Size (Cost) Too Big: Too small: • Requires too many resources • Won’t do the job Determining Sample Size for Mean pp.284-286 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 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 Obtaining Sample Size in PHStat PHStat | Sample Size | Determination for the mean… PHStat | Sample Size | Determination for the proportion… Here’s an Excel spreadsheet that contains the result of the previous examples: pp.290-291 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 Some Examples of Good and Bad Reporting Here is a link reporting Here is a link Here is a link practices Here is a link to a really terrible to a really bad practice to some not so bad to a better practice Summary Discussed Confidence Interval Estimation for the Mean ( Unknown) Addressed Determining Sample Size Discussed Confidence Interval Estimation and Ethical Issues