How do I know which distribution to use?

1. How do I know which distribution to use?

2. Three discrete distributions Proportional Poisson Binomial

3. Given a number of categories Probability proportional to number of opportunities Days of the week, months of the year Proportional Number of successes in n trials Have to know n, p under the null hypothesis Punnett square, many p=0.5 examples Binomial Number of events in interval of space or time n not fixed, not given p Car wrecks, flowers in a field Poisson

4. Proportional Binomial Binomial with large n, small p converges to the Poisson distribution Poisson

5. Examples: name that distribution • Asteroids hitting the moon per year • Babies born at night vs. during the day • Number of males in classes with 25 students • Number of snails in 1x1 m quadrats • Number of wins out of 50 in rock-paper-scissors

6. Proportional Generate expected values Calculate 2 test statistic Binomial Poisson

7. Sample Null hypothesis Test statistic Null distribution compare How unusual is this test statistic? P > 0.05 P < 0.05 Reject Ho Fail to reject Ho

8. Chi-squared goodness of fit test Null hypothesis: Data fit a particular Discrete distribution Sample Calculate expected values Chi-squared Test statistic • Null distribution: • 2 With N-1-p d.f. compare How unusual is this test statistic? P > 0.05 P < 0.05 Reject Ho Fail to reject Ho

9. The Normal Distribution

10. Babies are “normal”

11. Babies are “normal”

12. Normal distribution

13. Normal distribution • A continuous probability distribution • Describes a bell-shaped curve • Good approximation for many biological variables

14. Continuous Probability Distribution

15. The normal distribution is very common in nature Human body temperature Human birth weight Number of bristles on a Drosophila abdomen

16. Discrete probability distribution Continuous probability distribution Normal Poisson Probability Probability density Pr[X=2] = 0.22 Pr[X=2] = ? Probability that X is EXACTLY 2 is very very small

17. Discrete probability distribution Continuous probability distribution Normal Poisson Probability Probability density Pr[1.5≤X≤2.5] = Area under the curve Pr[X=2] = 0.22

18. Discrete probability distribution Continuous probability distribution Normal Poisson Probability Probability density Pr[1.5≤X≤2.5] = 0.06 Pr[X=2] = 0.22

19. Normal distribution Probability density x

20. Normal distribution Probability density x Area under the curve:

21. But don’t worry about this for now!

22. A normal distribution is fully described by its mean and variance

23. A normal distribution is symmetric around its mean Probability Density Y

24. About 2/3 of random draws from a normal distribution are within one standard deviation of the mean

25. About 95% of random draws from a normal distribution are within two standard deviations of the mean (Really, it’s 1.96 SD.)

26. Properties of a Normal Distribution • Fully described by its mean and variance • Symmetric around its mean • Mean = median = mode • 2/3 of randomly-drawn observations fall between - and + • 95% of randomly-drawn observations fall between -2 and +2

27. Standard normal distribution • A normal distribution with: • Mean of zero. (m = 0) • Standard deviation of one. (s = 1)

28. Standard normal table • Gives the probability of getting a random draw from a standard normal distribution greater than a given value

29. Standard normal table: Z = 1.96

30. Standard normal table: Z = 1.96 Pr[Z>1.96]=0.025

31. Normal Rules • Pr[X < x] =1- Pr[X > x] Pr[X<x] + Pr[X>x]=1 + = 1

32. Standard normal is symmetric, so... • Pr[X > x] = Pr[X < -x]

33. Normal Rules • Pr[X > x] = Pr[X < -x] • Pr[X < x] =1- Pr[X > x]

34. Sample standard normal calculations • Pr[Z > 1.09] • Pr[Z < -1.09] • Pr[Z > -1.75] • Pr[0.34 < Z < 2.52] • Pr[-1.00 < Z < 1.00]

35. What about other normal distributions? • All normal distributions are shaped alike, just with different means and variances

36. What about other normal distributions? • All normal distributions are shaped alike, just with different means and variances • Any normal distribution can be converted to a standard normal distribution, by Z-score

37. Z tells us how many standard deviations Y is from the mean The probability of getting a value greater than Y is the same as the probability of getting a value greater than Z from a standard normal distribution.

38. Z tells us how many standard deviations Y is from the mean Pr[Z > z] = Pr[Y > y]

39. Example: British spies MI5 says a man has to be shorter than 180.3 cm tall to be a spy. Mean height of British men is 177.0cm, with standard deviation 7.1cm, with a normal distribution. What proportion of British men are excluded from a career as a spy by this height criteria?

40. Draw a rough sketch of the question

41. m = 177.0cm s = 7.1cm y = 180.3 Pr[Y >y]

42. Part of the standard normal table Pr[Z > 0.46] = 0.32276, so Pr[Y > 180.3] = 0.32276

43. Sample problem MI5 says a woman has to be shorter than 172.7 cm tall to be a spy. The mean height of women in Britain is 163.3 cm, with standard deviation 6.4 cm. What fraction of women meet the height standard for application to MI5?