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STAT10x9.27.05BinomialDistribution

# STAT10x9.27.05BinomialDistribution - Midterm 1 Thursday...

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Midterm 1 Thursday, September 29 In class One Hour (but you’ll have until 2:25) Covers through Probability (chapter 15 in book) Closed note, closed book Any formulas needed will be provided Bring a calculator (doesn’t need to be fancy) Bring something to write with Probability Models for Count Data Bernoulli Random Variables Example : Evolution vs. Creationism. A Harris poll conducted in June ‘05 (sample size n=1000 people nationwide) found that 64% of American adults believed that humans were created directly by God and 36% believed that humans evolved or were the result of divinely guided evolution. A random person is sampled : let’s make a probability model for this person and define a random variable X Sample space S = {Created, Evolved} STAT 101-105 Introduction to Statistics 147

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This is called a Bernoulli Trial. If we stop many people at random , each person constitutes a separate, independent Bernoulli Trial. Bernoulli Trials - Three requirements 1. There are only two possible outcomes : success and failure 2. The probability of success p is identical for each trial 3. Trials are independent – the outcome of one trial does not affect the outcome of subsequent trials. Examples of Bernoulli Trials : Coin Flipping : {Heads, Tails}, P (Heads) = 0.5 STAT 101-105 Introduction to Statistics 148 Outcome Created Evolved Random Variable X 1 0 P (X=x) 0.64 0.36 Dan Bernoulli worked out the math for Bernoulli trials
Disease : {Infection, Non-infection}, P (Disease) = varies Polls : {Win Iraq, Loose Iraq}, P( Win) 0.37 Lottery : {Jackpot, No Jackpot}, P( Jackpot) 0 Example : Suppose that 60% or 72,000 of the 120,000 New Haven residents believe that God created humans. We sample one person who believes in creationism. The percentage of remaining residents that believe in creationism has shrunk : only 71,999/120,000 or 59.9997% believe in creationism.. For a large sample, no problem. BUT - What if - There are only 10 people in New Haven 60% (i.e. 6 people) believe in creationism We sample a creationist Remaining percentage of creationists is 5/9 or 54% These trials are no longer independent – the outcome of one trial changes the probability of success in subsequent trials. 10% Rule of Thumb : STAT 101-105 Introduction to Statistics 149

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Bernoulli Trials are reasonably independent as long as the sample number of trials is less than 10% of the population size. STAT 101-105 Introduction to Statistics 150
Binomial Random Variables Example : Seatbelt usage . Suppose that we decide to ask four people about their views on the origin of humans. Count the TOTAL number creationists out of the four sampled. What is the probability that exactly two are creationists? Might think : ( 29 ( 29 2 2 36 . 0 64 . 0 ts) creationis 4 of 2 ( = P However, this doesn’t account for order : there are six ways to observe two creationists among four people : each sequence is equally likely.

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