6_sampl_theory

6_sampl_theory - Last few slides from last time. Binomial...

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Last few slides from last time…
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Binomial distribution Distribution of number of successes in n independent trials. Probability of success on any given trial = p Probability of failure on any given trial = q = 1-p
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Mean and variance of a binomial random variable • The mean number of successes in a binomial experiment is given by: µ = np – n is the number of trials, p is the probability of success • The variance is given by σ 2 = npq –q = 1-p
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25 coin flips What is the probability that the number of heads is 14? We can calculate from the binomial formula that p(x 14) is .7878 (note this is not an approximation)
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Normal Approximation Using the normal approximation with µ = np = (25)(.5) = 12.5 and σ = sqrt(npq) = sqrt((25)(.5)(.5)) = 2.5 we get • p(x 14) = p(z (14-12.5)/2.5)) = p(z .6) = .7257 .7878 vs. .7257 -- not great!! Need a better approximation. ..
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Normal Approximation of Binomial Distribution 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 p(x) 0 1 2 3 4 5 6 7 8 9 1 01 11 21 31 41 51 61 71 81 92 02 12 22 32 42 5 Number of Successes
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Continuity Correction • Notice that the bars are centered on the numbers • This means that p(x 14) is actually the area under the bars less than x=14.5 • We need to account for the extra 0.5 •P ( x 14.5) = p(z .8) = .7881 -- a much better approximation! 9 1 01 11 21 31 41 51 61 7 Number of Successes
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Sampling theory 9.07 2/24/2004
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Goal for rest of today Parameters are characteristics of populations . – E.G. mean, variance We’ve also looked at statistics * of a sample. – E.G. sample mean, sample variance • How good are our statistics at estimating the parameters of the underlying population? * Statistic = a function of a sample. There are a lot of possible statistics, but some are more useful than others.
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Experimental design issues in sampling Suppose we want to try to predict the results of an election by taking a survey. Don’t want to ask everyone – that would be prohibitive! What percent of voters will vote for the Republican candidate for president? We ask 1000 eligible voters who they will vote for. In the next lecture, we will talk about estimating the population parameter (the % of voters who prefer the Republican) from the % of the survey respondents who say they favor him. But our statistics are only as good as our sampling technique – how do we pick those 1000 people for the survey?
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Simple random sample If the procedure for selecting n objects out of a large population of objects is such that all possible samples of n objects are equally likely, then we call the procedure a simple random sample . This is the gold standard for sampling. – Unbiased: each unit has the same probability of being chosen. – Independent: selection of one unit has no influence on selection of other units.
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In theory, how to get a simple random sample Get a list of every unit in the population.
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This note was uploaded on 11/11/2011 for the course BIO 9.07 taught by Professor Ruthrosenholtz during the Spring '04 term at MIT.

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6_sampl_theory - Last few slides from last time. Binomial...

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