Continuous Probability Distributions

1. Continuous Probability Distributions • Continuous random variable • Values from interval of numbers • Absence of gaps • Continuous probability distribution • Distribution of continuous random variable • Most important continuous probability distribution • The normal distribution

2. The Uniform Distribution • “Rectangular shaped” • Every value between a and b is equally likely • The mean and median are in the middle • Prob(X<=v) is the area on the left of v f(X) X a b v  Mean Median

3. The Normal Distribution • “Bell shaped” • Symmetrical • Mean, median and mode are equal • Interquartile rangeequals 1.33 s • 68-95-99 % rule • Random variablehas infinite range f(X) X  Mean Median

4. The Mathematical Model

5. Many Normal Distributions There are an infinite number of normal distributions By varying the parameters  and , we obtain different normal distributions

6. Finding Probabilities Probability is the area under the curve! f(X) X d c

7. Which Table to Use? An infinite number of normal distributions means an infinite number of tables to look up!

8. Solution: The Cumulative Standardized Normal Distribution Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .5478 .5000 0.0 .5040 .5080 Shaded Area Exaggerated .5398 .5438 .5478 0.1 0.2 .5793 .5832 .5871 Probabilities Z = 0.12 0.3 .6179 .6217 .6255 Only One Table is Needed

9. Standardizing Example Standardized Normal Distribution Normal Distribution Shaded Area Exaggerated

10. Example: Standardized Normal Distribution Normal Distribution Shaded Area Exaggerated

11. Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .5832 .5000 0.0 .5040 .5080 Shaded Area Exaggerated 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 Z = 0.21 0.3 .6179 .6217 .6255

12. Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .4168 .3821 -03 .3783 .3745 Shaded Area Exaggerated -02 .4207 .4168 .4129 -0.1 .4602 .4562 .4522 Z = -0.21 0.0 .5000 .4960 .4920

13. Example: Standardized Normal Distribution Normal Distribution Shaded Area Exaggerated

14. Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .6179 .5000 0.0 .5040 .5080 Shaded Area Exaggerated 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 Z = 0.30 0.3 .6179 .6217 .6255

15. Finding Z Values for Known Probabilities Cumulative Standardized Normal Distribution Table (Portion) What is Z Given Probability = 0.1217 ? .01 Z .00 0.2 0.0 .5040 .5000 .5080 .6217 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 .6179 .6255 .6217 0.3 Shaded Area Exaggerated

16. Recovering X Values for Known Probabilities Standardized Normal Distribution Normal Distribution

17. Finding Probabilities for X Values Using Excel Excel function: =NORMDIST(x,mean,standard_deviation,TRUE) =NORMSDIST(z,TRUE) Example Prob.(weight <= 165 lbs) when mean=180, std_dev=20: =NORMDIST(165,180,20,true) Answer: 0.2267 Prob.(weight >= 185 lbs) ? Prob.(weight <= 165 and weight <= 185 lbs) ?

18. Finding X Values for Known ProbabilitiesUsing Excel Excel function: =NORMINV(probabiltiy,mean,standard_deviation) =NORMSINV(probability) Example Prob.(weight <= X)= 0.2 (mean=180, std_dev=20) =NORMINV(0.2,180,20) Answer: X=163 Prob.(weight >= X)=0.4 X? Answer: X=185

19. Generating Random VariablesUsing Excel • Excel can be used to generate Discrete and Continuous Random Variables • Complex Probabilistic Models can be constructed and simulation can give insight and suggest managerial decisions • Tutorial

20. Assessing Normality • Not all continuous random variables are normally distributed • It is important to evaluate how well the data set seems to be adequately approximated by a normal distribution

21. Assessing Normality (continued) • Construct charts • For large data sets, does the histogram appear bell-shaped? • Compute descriptive summary measures • Do the mean, median and mode have similar values? • Is the interquartile range approximately 1.33 s? • Does the data obey the 68-95-99 percent rule? • Is the range approximately 6 s?

22. Assessing Normality (continued) • Observe the distribution of the data set • Do approximately 2/3 of the observations lie between mean 1 standard deviation? • Do approximately 4/5 of the observations lie between mean 1.28 standard deviations? • Do approximately 19/20 of the observations lie between mean 2 standard deviations?

23. Why Study Sampling Distributions • Sample statistics are used to estimate population parameters • e.g.: Estimates the population mean • Problems: different samples provide different estimate • Large samples gives better estimate; Large samples costs more • How good is the estimate? • Approach to solution: theoretical basis is sampling distribution

24. Sampling Distribution • Theoretical probability distribution of a sample statistic • Sample statistic is a random variable • Sample mean, sample proportion • Results from taking all possible samples of the same size

25. Example • Population: 100 subjects, numbered from 1 to 100 • Take sample of 10 and compute average • Take another sample, etc. • Excel workbook

26. Developing Sampling Distributions • Assume there is a population … • Population size N=4 • Random variable, X,is age of individuals • Values of X: 18, 20,22, 24 measured inyears C B D A

27. Developing Sampling Distributions (continued) Summary Measures for the Population Distribution P(X) .3 .2 .1 0 X A B C D (18) (20) (22) (24) Uniform Distribution

28. Developing Sampling Distributions All Possible Samples of Size n=2 (continued) 16 Sample Means 16 Samples Taken with Replacement

29. Developing Sampling Distributions Sampling Distribution of All Sample Means (continued) Sample Means Distribution 16 Sample Means P(X) .3 .2 .1 _ 0 X 18 19 20 21 22 23 24

30. Developing Sampling Distributions Summary Measures of Sampling Distribution (continued)

31. Comparing the Population with its Sampling Distribution Population N = 4 Sample Means Distribution n = 2 P(X) P(X) .3 .3 .2 .2 .1 .1 _ 0 0 X AB C D (18)(20)(22)(24) 18 19 20 21 22 23 24 X

32. Properties of Summary Measures • i.e. is unbiased • Standard error (standard deviation) of the sampling distribution is less than the standard error of other unbiased estimators • For sampling with replacement: • As n increases, decreases

33. Unbiasedness P(X) Unbiased Biased

34. Effect of Large Sample Larger sample size P(X) Smaller sample size

35. When the Population is Normal Population Distribution Central Tendency Variation Sampling Distributions Sampling with Replacement

36. When the Population is Not Normal Population Distribution Central Tendency Variation Sampling Distributions Sampling with Replacement

37. Central Limit Theorem the sampling distribution becomes almost normal regardless of shape of population As sample size gets large enough…

38. How Large is Large Enough? • For most distributions, n>30 • For fairly symmetric distributions, n>15 • For normal distribution, the sampling distribution of the mean is always normally distributed

39. Example: Standardized Normal Distribution Sampling Distribution

40. Population Proportions • Categorical variable • e.g.: Gender, voted for Bush, college degree • Proportion of population having a characteristic • Sample proportion provides an estimate • If two outcomes, X has a binomial distribution • Possess or do not possess characteristic

41. Sampling Distribution of Sample Proportion • Approximated by normal distribution • Mean: • Standard error: Sampling Distribution P(ps) .3 .2 .1 0 ps 0 . 2 .4 .6 8 1 p = population proportion

42. Standardizing Sampling Distribution of Proportion Standardized Normal Distribution Sampling Distribution

43. Example: Standardized Normal Distribution Sampling Distribution