Q1 Consider a Poisson random variable X with mean parameter = 5.

(a) Using rpois, generate n = 10 samples from this Poisson distribution and write down the sample observations.

(b) Calculate the index of dispersion based on your sample data and comment on its use with relevance to the Poisson distribution.

(c) Provide with reason an expression for the method of moment estimator of .

(d) Obtain the method of moment estimate of based on your sample.

(e) Estimate P(X = 6) based on your sample and exhibit your Rhas command. Provide the difference between your estimate and the true value of P(X = 6).

(f) Next obtain N = 100 bootstrap samples from your sample data and write down bootstrap means for first 5 bootstrap samples.

(g) Draw the histogram of bootstrap means based on those N = 100 bootstrap samples and comment on its relation with (a) the distribution of the method of moment estimator of and (b) the true mean parameter = 5.

(h) Describe clearly the procedure for constructing a bootstrap confidence interval for the mean height of the self-fertilised plants.Now obtain a 95% bootstrap confidence interval. Also, exhibit

your R-commands.

Q2 A paper in the Journal of Nervous and Mental Disorder reported the following observations related to the percentage of dextroamphetamine drug (7 hours after the administration) excreted by a sample of children having organically related disorders and a sample of children having

nonorganic disorders. Dextroamphetamine drug is used for treating hyperkinetic children.

Percentages related to organic disorders:

17.53, 20.60, 17.62, 28.93, 27.10.

Percentages related to non organic disorders:

5.59, 14.76, 13.32, 12.45, 12.79.

Let o and n denote the true mean percentage excretion corresponding to the organic and nonorganic disorders, respectively. We are interested to test the null hypothesis H0 : o = n against the alternative H1 :o > n.

(a) Explain why this is an unpaired data setup.

(b) Estimate the variances of the two groups and comment on the equality of the group variances.

(c) Use an appropriate parametric test at 5% level providing details of either the critical region or the p-value followed by your decision. You must state the underlying assumptions for implementing

parametric test and provide corresponding R-commands.

(d) Argue why a nonparametric test is preferable to the parametric test for this situation.

(e) Use an appropriate nonparametric test at 5% level for this testing problem providing details of either the critical region or the p-value followed by your decision. You must state the underlying assumptions for implementing nonparametric test and provide corresponding R-commands.