1. Confidence Intervals (Chapter 8) Confidence Intervals for numerical data: –Standard deviation known –Standard deviation unknown Confidence Intervals for categorical data

2. 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  ?

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

4. Point Estimates Estimate Population Parameters … with Sample Statistics Mean Proportion Variance Difference

5. 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

6. Confidence Interval Estimates Mean  Unknown Confidence Intervals Proportion  Known

7. 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

8. General Formula The general formula for a confidence interval is: 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 Point Estimate ± Margin of Error

9. 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

10. Level of Confidence Denoted by A relative frequency interpretation –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 –No probability involved in a specific interval

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

12. Factors Affecting Margin of error (Precision) Data variation –Measured by Sample size – Level of confidence – Intervals Extend from © 1984-1994 T/Maker Co. X - Z  to X + Z  xx

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

14. 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

15. 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.

16. 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 – Confidence Interval for ( Unknown)

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

18. Example

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

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

21. 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

22. Excel Tutorial Constructing Confidence Intervals using Excel: Tutorial Excel spreadsheet