1. How do I know which distribution to use?

2. Three discrete distributions
Proportional
Poisson
Binomial

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

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

18. Continuous probability distribution
Discrete probability
distribution
Continuous probability
distribution
Normal
Probability
Poisson
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]

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 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
x.x0
x.x1
x.x2
.x3
x.x4
x.x5
x.x6
x.x7
x.x8
x.x9
0.0
0.5
0.4721
0.1
0.4562
0.2
0.3
0.3707
0.4
0.3409
0.3336
0.2946
0.2776
Pr[Z > 0.46] = ,
so Pr[Y > 180.3] =

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