1. (a) Define the following terms: heteroscedasticity, parameter, test statistic, reference distribution. [4 marks]

   (b) What is a pivotal quantity? In general, how do you derive a test statistic from a pivotal quantity? In general, how do you derive a confidence interval from a pivotal quantity? [4 marks]

2. (a) Who discovered Student's t distribution? How did it come to be called that? [3 marks]

   (b) If x is a vector of observations, what does the following R code compute? [4 marks]

   > mu0 <- 27
   > 2*(1-pt(abs((mean(x)-mu0)/sqrt(var(x)/length(x))),length(x)-1))

3. 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) A fleet manager is testing two brands of tires. He randomly assigns pairs of tires, one of each brand, to the front wheels of 8 cars, randomly deciding which brand goes on the left or right wheel in each case, and runs the cars until the tires wear out. The results are shown below, in thousands of km.

   Car:         1     2     3     4     5     6     7     8
   Brand A:  36.9  45.3  36.2  32.1  37.2  48.4  38.2  33.5
   Brand B:  34.3  42.3  35.5  30.0  38.0  47.8  37.8  31.2

   (b) The deflection temperature under load for two different types of plastic pipe is being investigated. Two random samples of 10 pipe specimens were tested. The observed deflection temperatures are tabulated below, in degrees Fahrenheit.

   Type 1:  206  188  205  187  194  193  207  185  189  213
   Type 2:  177  197  206  201  180  176  185  200  197  192

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Statistics 3N03