## Description

1. Consider data collected by Brockman (1996) on female horseshoe crabs and the

number of male “satellites” residing near them. We will look at a subset of n = 41

of these female horseshoe crabs with the best spine condition. For this subset, the

numbers of female horseshoe crabs reporting particular numbers of satellites are as

shown in the table below.

Satellites (r) Frequency (fr)

0 19

1 3

2 1

3 4

4 7

5 7

Source: Brockman, H.J. (1996). Satellite Male Groups in Horseshoe Crabs, Limulus

polyphemus, Ethology 102 (1):1-21.

a. Assuming the number of satellites per female horseshoe crab follows a Poisson

distribution, estimate the mean number of satellites per female horseshoe crab.

b. Suppose we wish to test whether the distribution of the number of satellites

per female horseshoe crab is consistent with a Poisson distribution. Can a

chi-square goodness-of-fit test be applied to the data as presented in the table,

or do certain numbers of satellites need to be grouped?

If a grouping of numbers

of satellites is necessary, determine an appropriate grouping, showing evidence

that a chi-square goodness-of-fit test would indeed be appropriate for this

grouping.

c. Test whether the number of satellites per female horseshoe crab is consistent

with a Poisson distribution. Be sure to clearly state the null and alternative

hypotheses, present the test statistic and its distribution under the null hypothesis, and report the p-value and your conclusion at the α = 0.05 significance

level.

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2. Recall the dataset produced from a study carried out by the European CanCer

Organisation and analysed in Assignment 1. In that study, a non-invasive diagnostic

test for stomach and esophageal cancers was carried out on 335 people, and cancer

statuses and test results for these people were as shown in the table below.

Tested positive for stomach

Have stomach or or esophageal cancer?

esophageal cancer? No Yes

No 140 32

Yes 32 131

a. Using an odds ratio, describe and clearly interpret the association between

cancer status and test result.

b. Obtain a 95% confidence interval for the odds ratio θ calculated in part (a).

c. Is it appropriate to carry out a chi-square test of independence for the data

presented in the table? Briefly explain why or why not.

d. Regardless of your answer to part (c), carry out both Pearson and likelihood

ratio chi-square tests of independence to assess whether cancer status and test

result are associated. Be sure to clearly state the null and alternative hypotheses,

present the test statistic and its distribution under the null hypothesis, and

report the p-value and your conclusion at the α = 0.05 significance level.

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