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Unformatted text preview: STAT 200 Chapter 5 Sampling Distributions The binomial distribution for counts (Section 5.1) 1. The Binomial Experiment (a) The experiment consists of n identical trials. The number of trials is fixed in advance. (b) The outcomes on each trial can be dichotomized into two complementary cate gories: “success” and “failure”. (c) The probability of the success event, p , is the same from trial to trial. The probability of the failure event is 1 p . (d) The n trials are independent of each other. In other words, the results of previous trials do not affect the result of the current trial. Under the conditions of a binomial experiment, the random variable X representing the number/count of successes out of the n trials is a binomial random variable with parameters n and p . We write X ∼ Bin ( n,p ). 2. For X ∼ Bin ( n,p ), the probability that X will take on a value of k is given by P ( X = k ) = n k ! p k (1 p ) n k , k = 0 , 1 , 2 , ··· ,n where n k ! = n ! k !( n k )! is called the binomial coefficient (Another notation is n C k ). It gives the total number of ways (or combinations of successes and failures) of having X = k . Note that k ! = k × ( k 1) × ( k 2) × ... × 2 × 1 and 1! = 1, 0! = 1. 3. Mean (expected value), Variance and SD for a Binomial Random Variable X • Mean μ X : E ( X ) = n X k =0 k × P ( X = k ) = n X k =0 k × n k ! p k (1 p ) n k = np 1 • Variance σ 2 X : V ( X ) = n X k =0 ( k np ) 2 × P ( X = k ) = n X k =0 ( k np ) 2 × n k ! p k (1 p ) n k = np (1 p ) • Standard deviation σ X = SD ( X ) = q np (1 p ) • Interpretation of E ( X ): the average number of successes if you are to repeat the binomial experiment (each with n trials) many times. • Interpretation of V ( X ): a measure of the variability of the numbers of successes if the binomial experiment (each with n trials) is repeated many times. Sampling distribution of proportions (Section 5.1) • A local burger store is interested in finding out the proportion of vegetarian customers (who most likely purchase veggie burgers). It randomly samples 500 customers over a month and asks each whether he/she is vegetarian. Here, the population of interest are all customers visiting the burger store, and the 500 randomly chosen customers make up the sample. The parameter (a numerical summary of a population) is the proportion of all cus tomers who are vegetarian, and the statistic (a numerical summary of a sample) is the sample proportion of customers who are vegetarian. Sample data (of the burger store’s sample): Customer Vegetarian? 1 No 2 No 3 No 4 No 5 Yes : : : : 499 Yes 500 No 2 Suppose there are 32 vegetarian customers in the sample. The sample proportion of vegetarian customers is 32/500 = 0.064. How reliable is this sample proportion as an estimate of the true proportion of vegetarian customers?...
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 Spring '10
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 Binomial, Normal Distribution, Standard Deviation, Probability theory

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