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# StatsHomework2 - Stats 3003 Homework Set 2 1 Resampling a...

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Stats 3003 Homework Set 2 1. Resampling a Proportion For a binomial experiment, the proportion and its confidence interval can often be estimated easily, using de Moivre’s theorem. This analytical approach is fairly complicated, but we can do something simpler that’s just as effective; we can bootstrap. The idea is to draw a sample, with replacement, from our original sample, and get a “second opinion” estimate. We do this repeatedly (say 1000 times), and see how these resampling estimates vary. The /2 and 1 /2 quantiles of the resampled estimates give us a workable confidence interval. α −α Write an R program that calculates a bootstrap estimate of the success proportion for a sample of size n = 3000 where only x = 10 successes were observed, and calculate a 90% confidence interval (so = α 0.1) You can use this program as a guide: boot.p.hat <- function(s, reps=1000, alpha) { bp <- array(dim=reps) for (i in 1:reps) { } rs <- sample(s, length(s), replace=TRUE) bp[i] <- sum(rs)/length(rs) } hist(bp, main="Bootstrap Estimate of Proportion",freq=FALSE) lines(density(bp),col="blue") quantile(bp, c(alpha/2, 0.5, 1.0-alpha/2), type=9) s <- c(rep(1,45),rep(0,55)) # PUT YOUR DATA IN HERE boot.p.hat(s, 1000,0.05) For your answer, show the histogram, and your estimate and confidence interval in a simple paragraph. The Bootstrap part of the graph displays the ninety percent confidence level that was calculated in the equation. Both graphs are concurrent with each other and the data provided.

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2. The Birthday Problem, Revisited How many common birthdays in a large group? For a group of 730 students, use a Poisson approximation to find the estimated number of days where there are k common birth days, with k = 0, 1, . . . , 7. Show your results as a bar graph.
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StatsHomework2 - Stats 3003 Homework Set 2 1 Resampling a...

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