2. Lesoon. 4 - 1 Statistics for Management Confidence Interval Estimation

3. Lesoon. 4 - 2 Lesson Topics Confidence Interval Estimation for the Mean (  Known) Confidence Interval Estimation for the Mean (  Unknown) Confidence Interval Estimation for the Proportion The Situation of Finite Populations Sample Size Estimation

4. Lesoon. 4 - 3 Mean, , is unknown PopulationRandom Sample I am 95% confident that  is between 40 & 60. Mean X = 50 Estimation Process Sample

5. Lesoon. 4 - 4 Estimate Population Parameter... with Sample Statistic Mean  Proportion pp s Variances 2 Population Parameters Estimated  2 Difference  -  12 x - x 12 X _ __

6. Lesoon. 4 - 5 Provides Range of Values  Based on Observations from 1 Sample Gives Information about Closeness to Unknown Population Parameter Stated in terms of Probability Never 100% Sure Confidence Interval Estimation

7. Lesoon. 4 - 6 Confidence Interval Sample Statistic Confidence Limit (Lower) Confidence Limit (Upper) A Probability That the Population Parameter Falls Somewhere Within the Interval. Elements of Confidence Interval Estimation

8. Lesoon. 4 - 7 Parameter = Statistic ± Its Error © 1984-1994 T/Maker Co. Confidence Limits for Population Mean Error = Error = Error

9. Lesoon. 4 - 8 90% Samples 95% Samples  x _ Confidence Intervals 99% Samples X _

10. Lesoon. 4 - 9 Probability that the unknown population parameter falls within the interval Denoted (1 -  ) % = level of confidence e.g. 90%, 95%, 99%   Is Probability That the Parameter Is Not Within the Interval Level of Confidence

11. Lesoon. 4 - 10 Confidence Intervals Intervals Extend from (1 -  ) % of Intervals Contain .  % Do Not. 1 -   /2  X _  x _ Intervals & Level of Confidence Sampling Distribution of the Mean to

12. Lesoon. 4 - 11 Data Variation measured by  Sample Size Level of Confidence (1 -  ) Intervals Extend from © 1984-1994 T/Maker Co. Factors Affecting Interval Width X - Z  to X + Z  xx

13. Lesoon. 4 - 12 Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates

14. Lesoon. 4 - 13 Assumptions  Population Standard Deviation Is Known  Population Is Normally Distributed  If Not Normal, use large samples Confidence Interval Estimate Confidence Intervals (  Known)

15. Lesoon. 4 - 14 Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates

16. Lesoon. 4 - 15 Assumptions  Population Standard Deviation Is Unknown  Population Must Be Normally Distributed Use Student’s t Distribution Confidence Interval Estimate Confidence Intervals (  Unknown)

17. Lesoon. 4 - 16 Z t 0 t (df = 5) Standard Normal t (df = 13) Bell-Shaped Symmetric ‘Fatter’ Tails Student’s t Distribution

18. Lesoon. 4 - 17 Number of Observations that Are Free to Vary After Sample Mean Has Been Calculated Example  Mean of 3 Numbers Is 2 X 1 = 1 (or Any Number) X 2 = 2 (or Any Number) X 3 = 3 (Cannot Vary) Mean = 2 degrees of freedom = n -1 = 3 -1 = 2 Degrees of Freedom (df)

19. Lesoon. 4 - 18 Upper Tail Area df.25.10.05 11.0003.0786.314 2 0.8171.886 2.920 30.7651.6382.353 t 0 Assume: n = 3 df = n - 1 = 2  =.10  /2 =.05 2.920 t Values  / 2.05 Student’s t Table

20. Lesoon. 4 - 19 A random sample of n = 25 has = 50 and s = 8. Set up a 95% confidence interval estimate for .  .. 46695330 Example: Interval Estimation  Unknown

21. Lesoon. 4 - 20 Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates

22. Lesoon. 4 - 21 Assumptions  Sample Is Large Relative to Population n / N >.05 Use Finite Population Correction Factor Confidence Interval (Mean,  X Unknown) X  Estimation for Finite Populations

23. Lesoon. 4 - 22 Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates

24. Lesoon. 4 - 23 Assumptions  Two Categorical Outcomes  Population Follows Binomial Distribution  Normal Approximation Can Be Used  n·p  5 & n·(1 - p)  5 Confidence Interval Estimate Confidence Interval Estimate Proportion

25. Lesoon. 4 - 24 A random sample of 400 Voters showed 32 preferred Candidate A. Set up a 95% confidence interval estimate for p. p .053.107 Example: Estimating Proportion

26. Lesoon. 4 - 25 Sample Size Too Big: Requires too much resources Too Small: Won’t do the job

27. Lesoon. 4 - 26 What sample size is needed to be 90% confident of being correct within ± 5? A pilot study suggested that the standard deviation is 45. n Z Error    2 2 2 22 2 164545 5 2192220 .. Example: Sample Size for Mean Round Up

28. Lesoon. 4 - 27 What sample size is needed to be within ± 5 with 90% confidence? Out of a population of 1,000, we randomly selected 100 of which 30 were defective. Example: Sample Size for Proportion Round Up 228 

29. Lesoon. 4 - 28 What sample size is needed to be 90% confident of being correct within ± 5? Suppose the population size N = 500. Example: Sample Size for Mean Using fpc Round Up where 153 

30. Lesoon. 4 - 29 Lesson Summary Discussed Confidence Interval Estimation for the Mean(  Known) Discussed Confidence Interval Estimation for the Mean(  Unknown) Addressed Confidence Interval Estimation for theProportion Addressed the Situation of Finite Populations Determined Sample Size