MATH 503 Coursework 1
Submission deadline: 9 am Monday of Week 4 (1/11/2021).
ˆ You should answer both questions. The total marks is 20.
ˆ Solutions should be typed in a Word document and should be uploaded
via M503 Moodle in its designated CW1 space.
Alternatively, you can upload image of the handwritten solution. Any
handwritten solution should be clear and legible. The University ad-
vises that students use Microsoft Office Lens to record an image of their
solutions which can then be uploaded to Moodle.
ˆ Note that you are allowed to upload only one file in moodle.
ˆ There is no need to submit any R-code or a Rmarkdown file unless a
code has been specifically asked for in some questions like Qn 1(e).
ˆ Full marks will only be awarded for correct solutions which are pre-
sented in a clear manner with appropriate explanations
ˆ Order your solutions following the order of the questions and provide
page numbers. For example, solution to Qn 1(a) should be placed
before that to Qn 1(b) and so on.
ˆ Any graphs should be interpreted with a sentence or two to summarise
key findings.
ˆ Plagiarism: Please be aware of the difference between a general dis-
cussion on possible approaches to the coursework and working closely
together on detailed solutions. The latter is most certainly poor aca-
demic practice. You should implement and interpret all solutions on
your own. All writing should be your own. In particular, please do not
fall into the trap of ‘patchwork writing’: taking another person’s work
and restructuring their sentences by changing some of their words.
1
Q1 Consider a Poisson random variable X with mean parameter λ = 5.
(a) Using rpois, generate n = 10 samples from this Poisson distribu-
tion and write down the sample observations. [1]
(b) Calculate the index of dispersion based on your sample data and
comment on its use with relevance to the Poisson distribution. [2]
(c) Provide with reason an expression for the method of moment es-
timator of λ. [1]
(d) Obtain the method of moment estimate of λ based on your sam-
ple. [1]
(e) Estimate P (X = 6) based on your sample and exhibit your R-
command. Provide the difference between your estimate and the
true value of P (X = 6). [1]
(f) Next obtain N = 100 bootstrap samples from your sample data
and write down bootstrap means for first 5 bootstrap samples. [2]
(g) Draw the histogram of bootstrap means based on those N = 100
bootstrap samples and comment on its relation with (a) the dis-
tribution of the method of moment estimator of λ and (b) the true
mean parameter λ = 5. [2]
(h) Describe clearly the procedure for constructing a bootstrap confi-
dence interval for the mean of the Poisson distribution from which
you generated data for Part (a) of this question. Now obtain a 95%
bootstrap confidence interval. Also, exhibit your R-commands. [2]
(Q2 appears on the next page)
2
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 µo and µn denote the true mean percentage excretion corresponding
to the organic and nonorganic disorders, respectively. We are interested
to test the null hypothesis H0 : µo = µn against the alternative H1 :
µo > µn.
(a) Explain why this is an unpaired data setup. [1]
(b) Estimate the variances of the two groups and comment on the
equality of the group variances. [1]
(c) Use an appropriate parametric test at 5% level providing details
of either the critical region or the p-value followed by your deci-
sion. You must state the underlying assumptions for implementing
parametric test and provide corresponding R-commands. [2]
(d) Argue why a nonparametric test is preferable to the parametric
test for this situation. [2]
(e) Use an appropriate nonparametric test at 5% level for this test-
ing problem providing details of either the critical region or the
p-value followed by your decision. You must state the underly-
ing assumptions for implementing nonparametric test and provide
corresponding R-commands. [2]
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