1. (a) Define the following terms: heteroscedasticity, robustness, P-value, type I error, type II error. [5 marks]

   (b) What pivotal quantity is used to compare two variances? Under what assumptions is it valid? Write out the confidence interval formula derived from it, using Rosner's notation for the quantiles of the F distribution. [7 marks]
   
   (c) Give 5 interesting facts about William Sealy Gosset. [3 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) Students randomly assigned 22 giant freshwater prawns to two different aquaria and recorded their weight in grams after 25 days.

   Tank A: 0.85 0.94 0.82 1.19 0.52 1.14 0.76 1.59 1.11 0.66 0.76
   Tank B: 1.47 0.64 0.79 1.95 0.82 0.94 0.71 0.94 0.70 0.70 0.84

   (b) A random sample of 10 patients was treated with an anti-inflammatory cream on one arm (the left or right arm was chosen at random) and a placebo cream on the other arm. After 4 days a measure of muscle soreness was taken for each patient on each arm.

   Patient:  1  2  3  4  5  6  7  8  9 10
   Placebo: 46 22 10 14 26 29 29 47 20 13
   Treated:  2 32 30  3 14 32  2 39 18  2

3. Suppose that you are going to repeat the study described in 2(b) but this time you want the confidence interval for the mean difference to be ± 3 units. How many patients will you need? What would be achieved by doing the larger study? [5 marks]

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Statistics 2MA3