Sampling Distributions and Statistical Estimation Busstat 207

1. Sampling Distributions and Statistical EstimationBusstat 207 Shannon – Spring 2002

3. Sampling Error Concepts Exam Scores : Population

4. Sorted Population Parameters:

5. Population Distribution

6. Select Random Sample; n = 5 Sampling error = = 68.8 – 71.435

7. Select Random Sample; n = 5 Sampling error =

8. Select Random Sample; n = 5 Sampling error =

9. 200 Sample Means for n = 5

10. Sampling Error (200 samples, n = 5) Average Sampling Error = .50 Standard Deviation for Sampling Error = 7.83

12. Population Distribution

13. Select Random Sample; n = 10 Sampling error =

14. 200 Sample Means for n = 10

15. Sampling Error (200 samples, n = 10) Average Sampling Error = .32 Standard Deviation for Sampling Error = 5.16

18. Estimating Population Values • Point Estimation • Interval Estimation Darts and Horseshoes – the analogy

19. Confidence Intervals Lower Confidence Limit Upper Confidence Limit Point Estimate

20. 95% Confidence Intervals 0.95 z.025= -1.96 z.025= 1.96

21. Confidence Interval- General Format - Point Estimate  (Critical Value)(Standard Error)

22. Confidence Intervals The confidence level refers to a percentage greater than 50 and less than 100 that corresponds to the percentage of all possible confidence intervals, based on a given size sample, that will contain the true population value.

23. Confidence Intervals The confidence coefficient refers to the confidence level divided by 100% -- i.e., the decimal equivalent of a confidence level.

24. Confidence Interval- General Format:  known - Point Estimate  z (Standard Error)

25. Confidence Interval Estimates CONFIDENCE INTERVAL ESTIMATE FOR  ( KNOWN) where: z = Critical value from standard normal table  = Population standard deviation n = Sample size

26. Examples of a Confidence Interval Estimate for 

27. Special Message about Interpreting Confidence Intervals Once a confidence interval has been constructed, it will either contain the population mean or it will not. For a 95% confidence interval, if you were to produce all the possible confidence intervals using each possible sample mean from the population, 95% of these intervals would contain the population mean.

28. Margin of Error MARGIN OF ERROR (ESTIMATE FOR  WITH  KNOWN) where: e = Margin of error z = Critical value = Standard error of the sampling distribution

29. Confidence Interval Estimates CONFIDENCE INTERVAL ( UNKNOWN) where: t = Critical value from t-distribution with n-1 degrees of freedom = Sample mean s = Sample standard deviation n = Sample size

30. Confidence Interval Estimates CONFIDENCE INTERVAL-LARGE SAMPLE WITH  UNKNOWN where: z =Value from the standard normal distribution = Sample mean s = Sample standard deviation n = Sample size

31. Determining the Appropriate Sample Size SAMPLE SIZE REQUIREMENT - ESTIMATING  WITH  KNOWN where: z = Critical value for the specified confidence interval e = Desired margin of error  = Population standard deviation

32. Estimating A Population Proportion SAMPLE PROPORTION where: x = Number of occurrences n = Sample size

33. Estimating a Population Proportion STANDARD ERROR FOR p where:  =Population proportion n = Sample size

34. Confidence Interval Estimates for Proportions CONFIDENCE INTERVAL FOR  where: p = Sample proportion n = Sample size z = Critical value from the standard normal distribution

35. Examples of Confidence Interval for Proportion

36. Determining the Required Sample Size MARGIN OF ERROR FOR ESTIMATING where:  = Population proportion z = Critical value from standard normal distribution n = Sample size

37. Determining the Required Sample Size SAMPLE SIZE FOR ESTIMATING where:  = Value used to represent the population proportion e = Desired margin of error z = Critical value from the standard normal table

38. Confidence Coefficient Confidence Interval Confidence Level Degrees of Freedom Margin of Error Pilot Sample Point Estimate Sampling Error Student’s t-distribution Key Terms