Question

Although the sample sizes are not large for some of the examples below, you can use chi-square approximations to calculate p-values for the likelihood ratio tests.

(a) The following data are mean bacterial counts coming from four different methods for estimating the density of bacterial populations. 60.86 65.86 64.29 62.86 For each method, the number of observations giving the recorded mean was n 7. Assume the data for different plating methods are independent and that the observations for the jth method have a Poisson(Aj) distribution. The hypothesis of interest is whether the population means are the same, Ho : λ,-. λ4. Use a likelihood ratio test to test the hypothesis of interest. Show the steps leading to your conclusion. You can use without argum ent that the MLE for a random sample, x, , rn from a Poisson(A) distribution is λ i. (b) We showed in class that any (1 a) x 100% confidence region construction procedure can be converted to an a-level test. Show that the converse is also true. That is, show, that if φ60(x) is an a-level test of Ho : θ-6, then C(x)-(60 : so(x)-0) gives a (1-a) x 100% confidence region. Be sure to justify your argument.

Answer 1

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