1. HIM 3200 Normal Distribution Biostatistics Dr. Burton

2. Progression of a histogram into a continuous distribution -4 -3 -2 -1 0 1 2 3 4 z

3. Progression of a histogram into a continuous distribution -4 -3 -2 -1 0 1 2 3 4 z

4. Progression of a histogram into a continuous distribution -4 -3 -2 -1 0 1 2 3 4 z

5. Progression of a histogram into a continuous distribution -4 -3 -2 -1 0 1 2 3 4 z

6. Progression of a histogram into a continuous distribution -4 -3 -2 -1 0 1 2 3 4 z

7. Progression of a histogram into a continuous distribution -4 -3 -2 -1 0 1 2 3 4 z

8. Progression of a histogram into a continuous distribution -4 -3 -2 -1 0 1 2 3 4 z 0.4 0.3 0.2 0.1 0.0

9. Area under the curve -4 -3 -2 -1 0 1 2 3 4 0.4 0.3 0.2 0.1 0.0 = 50% 50%

10. Areas under the curve relating to z scores -4 -3 -2 -1 0 1 2 3 4 0.4 0.3 0.2 0.1 0.0 34.1% 0 to -1 34.1% 0 to +1

11. Areas under the curve relating to z scores -4 -3 -2 -1 0 1 2 3 4 0.4 0.3 0.2 0.1 0.0 68.2% -1 to -2+1 to +2 13.6%

12. Central limit theorem In reasonably large samples (25 or more) the distribution of the means of many samples is normal even though the data in individual samples may have skewness, kurtosis or unevenness. Therefore, a t-test may be computed on almost any set of continuous data, if the observations can be considered random and the sample size is reasonably large.

13. Areas under the curve relating to z scores -4 -3 -2 -1 0 1 2 3 4 0.4 0.3 0.2 0.1 0.0 68.2% 13.6% 95.4%

14. Areas under the curve relating to z scores -4 -3 -2 -1 0 1 2 3 4 0.4 0.3 0.2 0.1 0.0 95.4% 2.1% -2 to -3 +2 to +3

15. Areas under the curve relating to z scores -4 -3 -2 -1 0 1 2 3 4 0.4 0.3 0.2 0.1 0.0 99.6%

16. Areas under the curve relating to +z scores (one tailed tests) -4 -3 -2 -1 0 1 2 3 4 0.4 0.3 0.2 0.1 0.0 84.1% Acceptance area Critical area =15.9%

17. Areas under the curve relating to +z scores (one tailed tests) -4 -3 -2 -1 0 1 2 3 4 0.4 0.3 0.2 0.1 0.0 97.7% Acceptance area Critical area =2.3%

18. Areas under the curve relating to +z scores (one tailed tests) -4 -3 -2 -1 0 1 2 3 4 0.4 0.3 0.2 0.1 0.0 99.8% Acceptance area Critical area =0.2%

19. Asymmetric Distributions -4 -3 -2 -1 0 1 2 3 4 Positively Skewed Right Negatively Skewed Left

20. Distributions (Kurtosis) -4 -3 -2 -1 0 1 2 3 4 Flat curve = Higher level of deviation from the mean High curve = Smaller deviation from the mean

21. Distributions (Bimodal Curve) -4 -3 -2 -1 0 1 2 3 4

22. -3  -2  -- ++ +2  +3  -3 -2 0 123 Z scores Theoretical normal distribution with standard deviations Probability [% of area in the tail(s)] Upper tail.1587.02288.0013 Two-tailed.3173.0455.0027

23. What is the z score for 0.05 probability? (one-tailed test) 1.645 What is the z score for 0.05 probability? (two tailed test) 1.96 What is the z score for 0.01? (one-tail test) 2.326 What is the z score for 0.01 probability? (two tailed test) 2.576

24. The Relationship Between Z and X  =100  =15 X= Z=  Population Mean Standard Deviation 130 – 100 15 2

25. Central limit theorem In reasonably large samples (25 or more) the distribution of the means of many samples is normal even though the data in individual samples may have skewness, kurtosis or unevenness. Therefore, a t-test may be computed on almost any set of continuous data, if the observations can be considered random and the sample size is reasonably large.

26.  (x - x) 2 n - 1 s = Student’s t distribution t = x -  s / n Standard deviation

27. Standard Error of the Mean SE = s/ N 68 72 76 85 87 90 93 94 95 97 98 103 105 107 114 117 118 119 123 124 127 151 159 217 N = 15 X = 114.9 s = 34.1 s x = 8.8 Sample SE = 34.1/ 15 SE = 34.1/ 3.87 SE = 34.1/ 15 SE = 8.8  = 109.2  = 30.2

28. Confidence Intervals The sample mean is a point estimate of the population mean. With the additional information provided by the standard error of the mean, we can estimate the limits (interval) within which the true population mean probably lies. Source: Osborn

29. Confidence Intervals This is called the confidence interval which gives a range of values that might reasonably contain the true population mean The confidence interval is represented as:a    b –with a certain degree of confidence - usually 95% or 99% Source: Osborn

30. Confidence Intervals Before calculating the range of the interval, one must specify the desired probability that the interval will include the unknown population parameter - usually 95% or 99%. After determining the values for a and b, probability becomes confidence. The process has generated an interval that either does or does not contain the unknown population parameter; this is a confidence interval. Source: Osborn

31. Confidence Intervals To calculate the Confidence Interval (CI) Source: Osborn

32. Confidence Intervals In the formula,  is equal to 1.96 or 2.58 (from the standard normal distribution) depending on the level of confidence required: –CI 95,  = 1.96 –CI 99,  = 2.58 Source: Osborn

33. Confidence Intervals Given a mean of 114.9 and a standard error of 8.8, the CI 95 is calculated: = 114.9 + 17.248 = 97.7, 132.1 Source: Osborn )8.8(96.19.114 )/( 95   nsXCI 