Question

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7. Most of our problems will start off with a statement like “Let Xi, ,Xn be a random sample frorn the distributin However, this distributional assumption is often just that, an assumption. Sometimes that assumption is reasonable, other times it isn't. We can sometimes use simulation, together with known properties of the assumed distribution to check whether our distributional assumption is reasonable. To illustrate this, consider the following sample of bay anchovy larvae counts taken from the Hudson River: 158 143, 106, 57, 97, 80, 109, 109, 350, 224, 109, 214, 84 (a) If we assume that larvae are distributed randomly and uniformly in the river, then the number collected in a fixed size net could potentially follow a Poisson distribution. Explain why this scenario matches the characteristics of a Poisson distribution (b) To check whether or not the Poisson assumption is easonable, we can check whether the mean and variance of the observed data are consistent with the Poisson assumption. To start with, calculate the observed mean and variance of the data (e) If the data really do arise from a Poisson distribution, the mean and variance should be about the same (why?). In this case, they aren't. However, we wouldn't expect them to be exactly the same, even if the Poisson assumption is true. Why not? (d) The question remains, how should we expect Sto behve if the data really do arise from a Poisson distribution? To answer this question, we can carry out a simulation. We want our simulated data sets to mimic the one collected. So, we'll generate data sets of size n 13, since that was the sample size of the original data. In addition, well simulate the data sets assuming they come from a Poisson distribution with λ = since that's our best guess at the parameter. In your favorite simulation software (Excel works! So does R, SAS, Matlab, ), generate 100 data sets each with n = 13 observations. (e) For each data set, calculate S (f) Construct a histogram of your 100 simulated S2 values (g) Where does the s2 from the original data set fall on this histogram? How many of the simulated S2 values (h) What does this lead you to believe about the Poisson assumption? Is it reasonable in this case? What is your reasoning?

Answer 1

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