June 10, 2008Stat Lecture 9 - Proportions1 Introduction to Inference Sampling Distributions for Counts and Proportions Statistics Lecture 9

June 10, 2008Stat Lecture 9 - Proportions2 Administrative Notes Homework 3 is due on Monday, June 15 th –Covers chapters 1-5 in textbook Exam on Monday, June 15 th Review session on Thursday

June 10, 2008Stat Lecture 9 - Proportions3 Last Class Focused on models for continuous data: using the sample mean as our estimate of population mean Sampling Distributionof the Sample Mean how does the sample mean change over different samples? Population Parameter:  Distribution of these values? Sample 1 of size n x Sample 2 of size n x Sample 3 of size n x Sample 4 of size n x Sample 5 of size n x Sample 6 of size n x.

June 10, 2008Stat Lecture 9 - Proportions4 Today’s Class We will now focus on count data: categorical data that takes on only two different values “Success” (Y i = 1) or “Failure” (Y i = 0) Goal is to estimate population proportion: p = proportion of Y i = 1 in population

June 10, 2008Stat Lecture 9 - Proportions5 Examples Gender: our class has 83 women and 42 men What is proportion of women in Penn student population? Presidential Election: out of 2000 people sampled, 1150 will vote for McCain in upcoming election What proportion of total population will vote for McCain? Quality Control: Inspection of a sample of 100 microchips from a large shipment shows 10 failures What is proportion of failures in all shipments?

June 10, 2008Stat Lecture 9 - Proportions6 Inference for Count Data Goal for count data is to estimate the population proportion p From a sample of size n, we can calculate two statistics: 1. sample count Y 2. sample proportion = Y/n Use sample proportion as our estimate of population proportionp Sampling Distributionof the Sample Proportion how does sample proportion change over different samples? Population Parameter: p Distribution of these values? Sample 1 of size n x Sample 2 of size n x Sample 3 of size n x Sample 4 of size n x Sample 5 of size n x Sample 6 of size n x.

June 10, 2008Stat Lecture 9 - Proportions7 The Binomial Setting for Count Data 1.Fixed number n of observations (or trials) 2.Each observation is independent 3.Each observation falls into 1 of 2 categories: 1.Success (Y = 1) or Failure (Y = 0) 4.Each observation has the same probability of success: p = P(Y = 1)

June 10, 2008Stat Lecture 9 - Proportions8 Binomial Distribution for Sample Count Sample count Y (number of Y i =1 in sample of size n) has a Binomial distribution The binomial distribution has two parameters: number of trials n and population proportion p P(X=k) = nCk * p k (1-p) (n-k) Binomial formula accounts for number of success: p k number of failures : (1-p) n-k different orders of success/failures: nCk = n!/(k!(n-k)!)

June 10, 2008Stat Lecture 9 - Proportions9 Binomial Probability Histogram Can make histogram out of these probabilities Can add up bars of histogram to get any probability we want: eg. P(Y < 4) Different values of n and p have different histograms, but Table C in book has probabilities for many values of n and p

June 10, 2008Stat Lecture 9 - Proportions10 Binomial Table

June 10, 2008Stat Lecture 9 - Proportions11 Example: Genetics If a couple are both carriers of a certain disease, then their children each have probability 0.25 of being born with disease Suppose that the couple has 4 children P(none of their children have the disease)? P(X=0) = 4!/(0!*4!) *.25 0 * (1-.25) 4 P(at least two children have the disease)? P(Y ≥ 2) = P(Y = 2) +P(Y = 3) +P(Y = 4) = (from table) =

June 10, 2008Stat Lecture 9 - Proportions12 Example: Quality Control A worker inspects a sample of n=20 microchips from a large shipment The probability of a microchip being faulty is 10% (p = 0.10) What is the probability that there are less than three failures in the sample? P(Y < 3) = P(Y = 0) + P(Y =1) + P(Y = 2) = (from table) = 0.677

June 10, 2008Stat Lecture 9 - Proportions13 Sample Proportions Usually, we are more interested in a sample proportion = Y/n instead of a sample count P ( < k ) = P( Y < n*k) Example: a worker inspects a sample of 20 microchips from a large shipment with probability of a microchip being faulty is 0.1 What is the probability that our sample proportion of faulty chips is less than 0.05? P ( <.05 ) = P( Y < 1) = P(Y=0) = x 20

June 10, 2008Stat Lecture 9 - Proportions14 Mean and Variance of Binomial Counts If our sample count Y is a random variable with a Binomial distribution, what is the mean and variance of Y across all samples? Useful since we only observe the value of Y for our sample but what are the values in other samples? We can calculate the mean and variance of a Binomial distribution with parameters n and p: μ Y = n*p σ 2 = n*p*(1-p) σ = √ (n*p*(1-p))

June 10, 2008Stat Lecture 9 - Proportions15 Mean/Variance of Binomial Proportions Sample proportion is a linear transformation of the sample count ( = Y/n ) μ = 1/n * mean(Y) = 1/n * np = p Mean of sample proportion is true probability of success p σ 2 = 1/n 2 Var(Y) = 1/n 2 * n*p*(1-p) = p(1-p)/n Variance of sample proportion decreases as sample size n increases!

June 10, 2008Stat Lecture 9 - Proportions16 Variance over Long-Run Lower variance with larger sample size means that sample proportion will tend to be closer to population mean in larger samples Long-run behaviour of two different coin tossing runs. Much less likely to get unexpected events in larger samples

June 10, 2008Stat Lecture 9 - Proportions17 Binomial Probabilities in Large Samples In large samples, it is often tedious to calculate probabilities using the binomial distribution Example: Gallup poll for presidential election Bush has 49% of vote in population. What is the probability that Bush gets a count over 550 in a sample of 1000 people? P(Y > 550) = P(Y = 551) + P(Y = 552) + … + P(Y =1000) = 450 terms to look up in the table! We can instead use the fact that for large samples, the Binomial distribution is closely approximated by the Normal distribution

June 10, 2008Stat Lecture 9 - Proportions18

June 10, 2008Stat Lecture 9 - Proportions19 Normal Approximation to Binomial If count Y follows a binomial distribution with parameters n and p, then Y approximately follows a Normal distribution with mean and variance: μ Y = n*p This approximation is only good if n is “large enough”. Rule of thumb for “large enough”:n·p≥ 10 and n(1-p) ≥ 10 Also works for sample proportion: = Y/n follows a Normal distribution with mean and variance

June 10, 2008Stat Lecture 9 - Proportions20 Example: Quality Control Sample of 100 microchips (with usual 10% of microchips are faulty. What is the probability there are at least 17 bad chips in our sample? Using Binomial calculation/table is tedious. Instead use Normal approximation: Mean = n·p = 100  0.10 = 10 Var = n·p·(1-p) = 100  0.10  0.90 = 9 = P(Z ≥ 2.33) =1- P(Z ≤ 2.33) = 0.01 (from table)

June 10, 2008Stat Lecture 9 - Proportions21 Example: Gallup Poll Bush has 49% of vote in population What is the probability that Bush gets sample proportion over 0.51 in sample of size 1000? Use normal distribution with mean = p = 0.49 and variance p·(1-p)/n = = P(Z ≥1.27) =1- P(Z ≤1.27) = 0.102

June 10, 2008Stat Lecture 9 - Proportions22 Why does Normal Approximation work? Central Limit Theorem: in large samples, the distribution of the sample mean is approx. Normal Well, our count data takes on two different values: “Success” (Y i = 1) or “Failure” (Y i = 0) The sample proportion is the same as the sample mean for count data! So, Central Limit Theorem works for sample proportions as well!

June 10, 2008Stat Lecture 9 - Proportions23 Next Class - Lecture 10 Review session on Wednesday/Thursday –Show up with questions!