1. (a) Define the following terms: confidence interval, level of significance, type I error, type II error. [4 marks]
   
   (b) What applied problem motivated "Student" to study the Poisson distribution? [1 mark]

   (c) Give five interesting facts about R.A. Fisher. [5 marks]

2. Analyse the following two data sets with appropriate graphics and P-values. State your assumptions and your conclusions. Where possible, assess the validity of your assumptions. [30 marks]

   (a) The data below give the height (in cm) of plants, each grown with one of two different fertilizers, i.e., the fertilizer presently being used and a new fertilizer.

   Present:  48.2  54.6  58.3  47.8  51.4  52.0  55.2  49.1  49.9  52.6
   New:      52.3  57.4  55.6  53.2  61.3  58.0  59.8  54.8

   (b) To compare a new fertilizer with the fertilizer presently being used, 9 agricultural plots were chosen, each of the same area. Each plot was divided into two equal parts with the present fertilizer applied to one part (randomly chosen) and the new fertilizer to the other. The yield in kg/ha was measured in each case. To be cost-effective, the new fertilizer, being more expensive, would have to increase the mean yield by at least 250 kg/ha.

   Plot:        1    2    3    4    5    6    7    8    9
   New:      2250 2410 2260 2200 2360 2320 2240 2300 2090
   Present:  1920 2020 2060 1960 1960 2140 1980 1940 1790

3. Suppose that you are going to repeat the study described in 2(b) but this time you want the 95% confidence interval for the mean difference to be ± 10 units. How many plots will you need? Will this design be practical? [5 marks]

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Statistics 2MA3