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Confidence Intervals (Chapter 8) • Confidence Intervals for numerical data: – Standard deviation known – Standard deviation unknown • Confidence Intervals for categorical data Estimation Process: Example • We are interested in knowing the average household income in a certain county. • A sample with 144 observations yields a sample mean X=$72,000. • It is also “known” that in this county, =$24,000 • How can we get a “good” estimate for the true average household income ? Or: • How far away (“how bad”) can X be as an estimate for ? Estimation Process Population Random Sample Mean, , is unknown Mean X = 50 Sample I am 95% confident that is between 40 & 60. Point Estimates 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 General Formula The general formula for a confidence interval is: Point Estimate ± Margin of Error Point Estimate ± (Critical Value)(Standard Error) Where: • Point Estimate is the sample statistic estimating the population parameter of interest • Critical Value is a table value based on the sampling distribution of the point estimate and the desired confidence level • Standard Error is the standard deviation of the point estimate Elements of Confidence Interval Estimation • Level of confidence – Confidence in which the interval will contain the unknown population parameter • Precision (range) – 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 Margin of error (Precision) • Data variation – Measured by • Sample size – 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 Do You Ever Truly Know σ? • Probably not! • In virtually all real world business situations, σ is not known. • If there is a situation where σ is known then µ is also known (since to calculate σ you need to know µ.) • If you truly know µ there would be no need to gather a sample to estimate it. 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) 0 Z 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 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 1 pS pS 1 pS pS Z / 2 p pS Z / 2 n 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 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 Excel Tutorial Constructing Confidence Intervals using Excel: • Tutorial •Excel spreadsheet