HLTH 300 Biostatistics for Public Health Practice, Raul Cruz-Cano, Ph.D

1. Fox/Levin/Forde, Elementary Statistics in Social Research, 12e • Chapter 5: Probability and the Normal Curve HLTH 300 Biostatistics for Public Health Practice,Raul Cruz-Cano, Ph.D 2/24/2014, Spring 2014

2. Announcements • Okay, let’s have a review before the exams…but we need to reduce the number of exams • (4 Exams+ 4 Reviews = 8 Sessions = .5 of our meetings!) • See new syllabus • Rule for the reviews: • Specific questions, not “blanket” questions. • Not obligatory to attend. • For the exams you are allowed to bring copies of Appendix C and Appendix D

3. CHAPTER OBJECTIVES 5.1 • Calculate probabilities and understand the rules of probability 5.2 • Understand the concept of a probability distribution 5.3 • List the characteristics of the normal curve 5.4 • Understand the area under the normal curve 5.5 • Calculate and use z scores

4. Learning Objectives • After this lecture, you should be able to complete the following Learning Outcomes • 5.1 Calculate probabilities and understand the rules of probability

5. 5.1 Probability The relative likelihood of occurrence of any given outcome P = 0 The outcome is Impossible P = .5 The outcome is as likely to happen as not happen P = 1 The outcome is certain

6. 5.1 The Rules of Probability • Converse Rule: The probability that something will not occur • Addition Rule: The probability of obtaining one of several different and distinct outcomes (mutually exclusive) • Multiplication Rule: The probability of obtaining two or more outcomes in combination (Independent)

7. 5.1 The Rules of Probability • Addition Rule: The probability of obtaining one of several different outcomes (not mutually exclusive) -P (A and B)

8. Example A= Intoxicated with C02 B= Intoxicated with N02 What is P(A)? P(B)? What is P(Ā)? What is P(A and B)? What is P(A or B)

9. 5.1 Probability: Example Heads or Tails? Let’s flip a coin two times: Probability of heads on the first flip: Probability of heads on the second flip: Probability of getting heads on both flip:

10. 5.1 Probability: Example Heads or Tails? What is the probability that one flip in two flips will land on heads?

11. Example • In the U.S. the probability of a driver being uninsured is .123 • Two drivers crash: • P(Both uninsured) • P(Both Insured) • Five Drivers Crash, P(all uninsured)?

12. Learning Objectives • After this lecture, you should be able to complete the following Learning Outcomes • 5.2 Understand the concept of a probability distribution

13. 5.2 Probability Distributions Directly analogous to a frequency distribution • Except it is based on probability theory • Standard Deviation = σ Mean = μ

14. 5.2 Figure 5.1

15. Examples Problem 6: Standard 6-sided die: X: Outcome of roll P(X=2) P(X=3 or X=4) P(X=Odd Number) P(X=Anything but 5)

16. Examples Problem 14 X: A random politician from the sample P(X=Republican) P(X=Democrat that support euthanasia) P(X=Does not support euthanasia) P(X=Republican that does not support euthanasia)

17. Examples Problem 20 Lottery ticket 2 numbers and a letter, e.g. 3 7 P P(match 1st digit) P(match 2nd digit) P(Not match 1st digit) P(match 1st and 2nd digit) P(match letter) P(Perfect match)

18. Learning Objectives • After this lecture, you should be able to complete the following Learning Outcomes • 5.3 List the characteristics of the normal curve

19. 5.3 Characteristics of the Normal Curve • Smooth • Symmetrical • Unimodal • Mean = Median = Mode • Infinite in Both Directions • Probability Distribution • Mean = μ; Standard Deviation = σ • Areas Under the Curve = 100%

20. Characteristics of the Normal Curve If you know the mean and the std. deviation of a normally distributed variable then you can find many probabilities: P(X < x ) P(X > x ) P(x1 <X < x2) If you know the mean and the std. deviation of the grades of exams for a class (normally distributed) then you can find many probabilities: X: Grade of a random student P(X < 90 ) P(X > 85 ) P(75 < X < 95

21. 5.3 The Reality of the Normal Curve The normal curve is a theoretical ideal Many many variables do conform to the normal curve… few students get low grades, few students get great grades, most get around average Some variables do not conform to the normal curve • Many distributions are skewed, multi-modal, and symmetrical but not bell-shaped • Assuming normality when it does not exist can impact the validity of our conclusions

22. Learning Objectives • After this lecture, you should be able to complete the following Learning Outcomes • 5.4 Understand the area under the normal curve

23. 5.4 Figure5.5 Like a very smooth histogram

24. 5.4 Figure 5.6

25. 5.4 Figure 5.7

26. 5.4 Figure 5.8

27. Learning Objectives • After this lecture, you should be able to complete the following Learning Outcomes • 5.5 Calculate and use z scores

28. 5.5 Standard Scores and the Normal Curve It is possible to determine the area under the curve for any sigma distance from the mean This distance is called a z-score • Indicates direction and distance that any raw score deviates from the mean in sigma units

29. Example X: Salaries in a company Mean= $20,000 Std. Dev=$1,500 Distribution = Normal P(20,000 < X < 22,000)=?

30. P(20,000 < X < 22,000) =40.82% Exercises: P(X < 22,000) P(X < 22,000) P(18,000 < X < 22,000) P(X<18,000) P(X>18,000)

31. 5.5 Finding Probability under the Normal Curve When the normal curve is used in conjunction with z scores and Table A in Appendix C, we can determine the probability of obtaining any raw score (X) in a distribution • The converse, addition, and multiplication rules still apply We can also reverse this process to calculate score values from particular portions of area or percentages

32. X:911 Response time Mean= 5.6 minutes Std. Dev= 1.8 minutes Distribution = Normal P(X<x)=75%

34. Examples Problem 27 X: SAT Scores Normally distributed Mean=500 Std. Dev. = 100 P(500< X <600) P(400 < X < 600) e. P(X>600) f. P(X<300)

35. Homework Chapter 5: Problems 13, 16 and 30

36. CHAPTER SUMMARY • Probabilities can be calculated using the converse, addition, and multiplication rules 5.1 • The probability distribution is analogous to a frequency distribution and includes a mean and standard deviation 5.2 • The normal curve is a theoretical ideal and therefore cannot be applied to all distributions 5.3 • 100% of the data falls under the normal curve, with 50% of the data falling to either side of the mean 5.4 • By converting raw scores to z scores, we can determine the probability of randomly selecting an individual with that score from the population 5.5