The Breeding Birds Survey is organized by the British Trust for Ornithology to monitor the relative abundance of different species of birds over time and in different areas of the UK. Each year, volunteers survey kilometre square sites at regular intervals, and note down the numbers of birds of each species that they observe on each occasion. In 2001, access to the countryside was restricted by a national outbreak of foot and mouth disease.

The SPSS file rails.sav contains data on counts of coots and moorhens collected over the period 1994-2007. There are four variables: year, sites, coot and moorhen.

(a) The variable sites gives the total number of sites surveyed each year. How did the restricted access in 2001 due to the national outbreak of foot and mouth disease affect the number of sites surveyed? [1]

(b) The variables coot and moorhen contain, respectively, the annual densities of coots and moorhens per square kilometre, over the survey sites. Obtain a multiple line plot of coot and moorhen by year (but do not include it in your answer). Edit this multiple line plot by adding a suitable title and changing the label for the vertical axis to something more informative than the default label Value. Include a copy of this edited graph with your answer. Comment briefly on the relative densities of coots and moorhens, and how they have changed over time. [4]

(c) For each of the following, state whether it is a discrete or continuous random variable: sites and coot. [1]

(d) Obtain a scatterplot of coot against moorhen, and include a copy of the graph with your answer. Comment briefly on the relationship, if any, between the two variables. [2]

(e) Obtain the correlation coefficient between moorhen and coot, and obtain the significance probability for the test of the null hypothesis that the two variables are uncorrelated. What do you conclude from this test? [2]

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(f) Create a new variable diff, containing the differences between the annual densities of coots and moorhens. (Part (a) of Exercise 7.1 of Introduction to statistical modelling is similar to this.) Obtain a histogram of diff, customized so that the bin size is 0.01. Include a copy of your final histogram with your answer. [2]

(g) Obtain the mean and standard deviation of diff, and a 95% confidence interval for the mean. (You may assume that any conditions required for using the confidence interval are satisfied.) Is it plausible that the underlying mean densities of coots and moorhens are the same? Explain your reasoning.