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# Lesson04_Static11

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### Lesson04_Static11

1. 1. Statistics for Management Confidence Interval Estimation
2. 2. Lesson Topics <ul><li>Confidence Interval Estimation for the Mean </li></ul><ul><li>(  Known) </li></ul><ul><li>Confidence Interval Estimation for the Mean </li></ul><ul><li> (  Unknown) </li></ul><ul><li>Confidence Interval Estimation for the </li></ul><ul><li>Proportion </li></ul><ul><li>The Situation of Finite Populations </li></ul><ul><li>Sample Size Estimation </li></ul>
3. 3. Mean,  , is unknown Population Random Sample I am 95% confident that  is between 40 & 60. Mean X = 50 Estimation Process Sample
4. 4. Estimate Population Parameter... with Sample Statistic Mean  Proportion p p s Variance s 2 Population Parameters Estimated  2 Difference  -  1 2 x - x 1 2 X _ _ _
5. 5. <ul><li>Provides Range of Values </li></ul><ul><ul><li>Based on Observations from 1 Sample </li></ul></ul><ul><li>Gives Information about Closeness to Unknown Population Parameter </li></ul><ul><li>Stated in terms of Probability </li></ul><ul><ul><li>Never 100% Sure </li></ul></ul>Confidence Interval Estimation
6. 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
7. 7. Parameter = Statistic ± Its Error © 1984-1994 T/Maker Co. Confidence Limits for Population Mean Error = Error = Error Error
8. 8. 90% Samples 95% Samples Confidence Intervals 99% Samples X _  x _
9. 9. <ul><li>Probability that the unknown </li></ul><ul><li>population parameter falls within the </li></ul><ul><li> interval </li></ul><ul><li>Denoted (1 -  ) % = level of confidence e.g. 90%, 95%, 99% </li></ul><ul><ul><li> Is Probability That the Parameter Is Not Within the Interval </li></ul></ul>Level of Confidence
10. 10. Confidence Intervals Intervals Extend from (1 -  ) % of Intervals Contain  .   % Do Not. 1 -   /2  /2 X _  x _ Intervals & Level of Confidence Sampling Distribution of the Mean to
11. 11. <ul><li>Data Variation </li></ul><ul><li>measured by  </li></ul><ul><li>Sample Size </li></ul><ul><li>Level of Confidence (1 -  ) </li></ul>Intervals Extend from © 1984-1994 T/Maker Co. Factors Affecting Interval Width X - Z  to X + Z  x x
12. 12. Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates
13. 13. <ul><li>Assumptions </li></ul><ul><ul><li>Population Standard Deviation Is Known </li></ul></ul><ul><ul><li>Population Is Normally Distributed </li></ul></ul><ul><ul><li>If Not Normal, use large samples </li></ul></ul><ul><li>Confidence Interval Estimate </li></ul>Confidence Intervals (  Known)
14. 14. Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates
15. 15. <ul><li>Assumptions </li></ul><ul><ul><li>Population Standard Deviation Is Unknown </li></ul></ul><ul><ul><li>Population Must Be Normally Distributed </li></ul></ul><ul><li>Use Student’s t Distribution </li></ul><ul><li>Confidence Interval Estimate </li></ul>Confidence Intervals (  Unknown)
16. 16. Z t 0 t ( df = 5) Standard Normal t ( df = 13) Bell-Shaped Symmetric ‘ Fatter’ Tails Student’s t Distribution
17. 17. <ul><li>Number of Observations that Are Free </li></ul><ul><li> to Vary After Sample Mean Has Been </li></ul><ul><li> Calculated </li></ul><ul><li>Example </li></ul><ul><ul><li>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 </li></ul></ul>degrees of freedom = n -1 = 3 -1 = 2 Degrees of Freedom ( df )
18. 18. Upper Tail Area df .25 .10 .05 1 1.000 3.078 6.314 2 0.817 1.886 2.920 3 0.765 1.638 2.353 t 0 Assume: n = 3 df = n - 1 = 2   = .10  /2 =.05 2.920 t Values  / 2 .05 Student’s t Table
19. 19. <ul><li>A random sample of n = 25 has = 50 and </li></ul><ul><li>s = 8. Set up a 95% confidence interval estimate for  . </li></ul>   . . 46 69 53 30 Example: Interval Estimation  Unknown
20. 20. Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates
21. 21. <ul><li>Assumptions </li></ul><ul><ul><li>Sample Is Large Relative to Population </li></ul></ul><ul><ul><ul><li>n / N > .05 </li></ul></ul></ul><ul><li>Use Finite Population Correction Factor </li></ul><ul><li>Confidence Interval (Mean,  X Unknown) </li></ul>X    Estimation for Finite Populations
22. 22. Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates
23. 23. <ul><li>Assumptions </li></ul><ul><ul><li>Two Categorical Outcomes </li></ul></ul><ul><ul><li>Population Follows Binomial Distribution </li></ul></ul><ul><ul><li>Normal Approximation Can Be Used </li></ul></ul><ul><ul><li>n · p  5 & n· (1 - p )  5 </li></ul></ul><ul><li>Confidence Interval Estimate </li></ul>Confidence Interval Estimate Proportion
24. 24. <ul><li>A random sample of 400 Voters showed 32 preferred Candidate A. Set up a 95% confidence interval estimate for p . </li></ul>p   .053 .107 Example: Estimating Proportion
25. 25. Sample Size <ul><li>Too Big: </li></ul><ul><li>Requires too </li></ul><ul><li>much resources </li></ul><ul><li>Too Small: </li></ul><ul><li>Won’t do </li></ul><ul><li>the job </li></ul>
26. 26. <ul><li>What sample size is needed to be 90% confident of being correct within ± 5 ? A pilot study suggested that the standard deviation is 45. </li></ul>n Z Error     2 2 2 2 2 2 1 645 45 5 219 2 220  . . Example: Sample Size for Mean Round Up
27. 27. <ul><li>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. </li></ul>Example: Sample Size for Proportion Round Up 228 
28. 28. <ul><li>What sample size is needed to be 90% confident of being correct within ± 5 ? Suppose the population size N = 500. </li></ul>Example: Sample Size for Mean Using fpc Round Up where 153 
29. 29. Lesson Summary <ul><li>Discussed Confidence Interval Estimation for </li></ul><ul><li> the Mean (  Known) </li></ul><ul><li>Discussed Confidence Interval Estimation for </li></ul><ul><li> the Mean (  Unknown) </li></ul><ul><li>Addressed Confidence Interval Estimation for </li></ul><ul><li> the Proportion </li></ul><ul><li>Addressed the Situation of Finite Populations </li></ul><ul><li>Determined Sample Size </li></ul>