Proportions

First, some more probability! Why were there so many birthday pairs? 14 people 91 pairs

Permutation An ordered set of events Boy, Girl Girl, Boy

Combination An unordered set of events Boy, Girl Girl, Boyor

Permutations - Example If you have five children, three boys and two girls, how many possible birth orders are there? Example: BGBGB

Checking... BBBGGBBGBGBBGGBBGBBGBGBGB BGGBBGBBBGGBBGBGBGBBGGBBB Answer = 10 permutations One combination (three boys, two girls)

0!=1 1!=1

Multiplication rule If two events A and B are independent, then Pr[A and B] = Pr[A] x Pr[B]

Permutations and probabilities The multiplication rule also holds for permutations As long as the events are independent!

Pr[BBBGG]=Pr[B]*Pr[B]*Pr[B]*Pr[G]*Pr[G] =0.512*0.512*0.512*0.488*0.488 =(0.512) 3 *(0.488) 2

Probability of a combination Consider the previous example: What is the probability of the combination three boys, two girls?

Probability of a combination = (Number of permutations giving that combination)  (Probability of one of those permutations) * Assumes that events are independent

Probability of a combination =10  * Assumes that events are independent = 0.31

Generalizing… =  p x (1-p) n-x * Assumes that events are independent

The probability of a combination of independent events

The binomial distribution

Binomial distribution The probability distribution for the number of “successes” in a fixed number of independent trials, when the probability of success is the same in each trial

The binomial distribution: P(x) = probability of a total of x successes p = probability of success in each trial n = total number of trials

Sheep null distribution from computer simulation

Binomial distribution, n = 20, p = 0.5

Remember:

Example: Paradise flycatchers

A population of paradise flycatchers has 80% brown males And 20% white. You capture 5 male flycatchers at random. What is the probability that 3 of these are brown and 2 are white?

Example: Paradise flycatchers A population of paradise flycatchers has 80% brown males And 20% white. You capture 5 male flycatchers at random. What is the probability that 3 of these are brown and 2 are white? n = 5x = 3p = 0.8

Example: Paradise flycatchers A population of paradise flycatchers has 80% brown males And 20% white. You capture 5 male flycatchers at random. What is the probability that 3 of these are brown and 2 are white? n = 5x = 3p = 0.8

Example: Paradise flycatchers A population of paradise flycatchers has 80% brown males And 20% white. You capture 5 male flycatchers at random. What is the probability that 3 of these are brown and 2 are white? n = 5x = 3p = 0.8

Mean: Variance: Properties of the binomial distribution

Proportion of successes in a sample

Example: A coin is flipped 8 times; 5 were heads Proportion of heads = 5/8

Properties of proportions Mean = p Variance = Standard error =

Sampling distribution of a proportion

The law of large numbers The greater the sample size, the closer an estimate of a proportion is likely to be to its true value. Sample size

Estimating a proportion

Confidence interval for a proportion* where Z = 1.96 for a 95% confidence interval * The Agresti-Couli method

Example: The daughters of radiologists 30 out of 87 offspring of radiologists are males, and the rest female. What is the best estimate of the proportion of sons among radiologists?

Hypothesis testing on proportions The binomial test

Binomial test

Example

Hypotheses

N =10, p 0 =0.2

P =0.12 The is greater than the common  value of 0.05, so we would not reject the null hypothesis.

Summary