1. Normal Distributions Z Transformations Central Limit Theorem Standard Normal Distribution Z Distribution Table Confidence Intervals Levels of Significance Critical Values Population Parameter Estimations

2. Normal Distribution

3. Mean 

4. Normal Distribution Mean  Variance  2

5. Normal Distribution Mean  Variance  2 Standard Deviation 

6. Normal Distribution Mean  Variance  2 Standard Deviation  Z Transformation

7. Normal Distribution Mean  Variance  2 Standard Deviation  Pick any point X along the abscissa.

8. Normal Distribution Mean  Variance  2 Standard Deviation  x

9. Normal Distribution Mean  Variance  2 Standard Deviation  x Measure the distance from x to .

10. Normal Distribution Mean  Variance  2 Standard Deviation   x –  x Measure the distance from x to .

11. Normal Distribution Mean  Variance  2 Standard Deviation  Measure the distance using z as a scale; where z = the number of  ’s.  x

12. Normal Distribution Mean  Variance  2 Standard Deviation  Measure the distance using z as a scale; where z = the number of  ’s.  x zz

13. Normal Distribution Mean  Variance  2 Standard Deviation   x –  zz x Both values represent the same distance.

14. Normal Distribution Mean  Variance  2 Standard Deviation   x x –  = z 

15. Normal Distribution Mean  Variance  2 Standard Deviation   x x –  = z  z = (x –  ) / 

16. Z Transformation for Normal Distribution Z = ( x –  ) / 

17. Central Limit Theorem The distribution of all sample means of sample size n from a Normal Distribution ( ,  2 ) is a normally distributed with Mean =  Variance =  2 / n Standard Error =  / √n

18. Sampling Normal Distribution Sample Size n Mean  Variance  2 / n Standard Error  / √n 

19. Sampling Normal Distribution Sample Size n Mean  Variance  2 / n Standard Error  / √n Pick any point X along the abscissa.  x

20. Sampling Normal Distribution Sample Size n Mean  Variance  2 / n Standard Error  / √n z = ( x –  ) / (  / √n)  x

21. Z Transformation for Sampling Distribution Z = ( x –  ) / (  / √n)

22. Standard Normal Distribution & The Z Distribution Table What is a Standard Normal Distribution?

23. Standard Normal Distribution Mean  = 0

24. Standard Normal Distribution Mean  = 0 Variance  2 = 1

25. Standard Normal Distribution Mean  = 0 Variance  2 = 1 Standard Deviation  = 1

26. Standard Normal Distribution Mean  = 0 Variance  2 = 1 Standard Deviation  = 1 What is the Z Distribution Table?

27. Z Distribution Table The Z Distribution Table is a numeric tabulation of the Cumulative Probability Values of the Standard Normal Distribution.

28. Z Distribution Table The Z Distribution Table is a numeric tabulation of the Cumulative Probability Values of the Standard Normal Distribution. What is “Z” ?

29. Define Z as the number of standard deviations along the abscissa. Practically speaking, Z ranges from -4.00 to +4.00  (-4.00) = 0.00003 and  (+4.00) = 0.99997

30. Standard Normal Distribution Mean  = 0 Variance  2 = 1 Standard Deviation  = 1 Area under the curve = 100% z = -4.00 z = +4.00

31. Normal Distribution Mean  Variance  2 Standard Deviation  Area under the curve = 100% z = -4.00z = +4.00 And the same holds true for any Normal Distribution !

32. Sampling Normal Distribution Sample Size n Mean  Variance  2 / n Standard Error  / √n Area = 100% As well as Sampling Distributions ! z = -4.00 z = +4.00

33. Confidence Intervals Levels of Significance Critical Values

34. Confidence Intervals Example: Select the middle 95% of the area under a normal distribution curve.

35. Confidence Interval 95% 95%

36. Confidence Interval 95% 95% 95% of all the data points are within the 95% Confidence Interval

37. Confidence Interval 95% 95% Level of Significance  = 100% - Confidence Interval

38. Confidence Interval 95% 95% Level of Significance  = 100% - Confidence Interval  = 100% - 95% = 5%

39. Confidence Interval 95% 95% Level of Significance  = 100% - Confidence Interval  = 100% - 95% = 5%  /2 = 2.5%

40.  / 2  5% Confidence Interval 95% Level of Significance  5%

41.  / 2  5% Confidence Interval 95% Level of Significance  5% From the Z Distribution Table For  (z) = 0.025 z = -1.96 And  (z) = 0.975 z = +1.96

42.  / 2  5% Confidence Interval 95% Level of Significance  5%      

43. Calculating X Critical Values X critical values are the lower and upper bounds of the samples means for a given confidence interval. For the 95% Confidence Interval X lower = (  - X) Z  /2 / ( s / √n) where Z  /2 = -1.96 X upper = (  - X) Z  /2 / ( s / √n) where Z  /2 = +1.96

44.  / 2  5% Confidence Interval 95% Level of Significance  5%       X lower X upper

45. Estimating Population Parameters Using Sample Data

46. A very robust estimate for the population variance is  2 = s 2. A Point Estimate for the population mean is  = X. Add a Margin of Error about the Mean by including a Confidence Interval about the point estimate. FromZ = ( X –  ) / (  / √n)  = X ± Z  /2 (s / √n) For 95%, Z  /2 = ±1.96