1. (a) Define the following terms: homoscedasticity, type I error, type II error, power. [4 marks]

   (b) Describe an application where the pivotal quantity (n-1)s²/s² ~ c²(n-1) could be used. [3 marks]  

2. What useful result does the following R code demonstrate by simulation? [3 marks]

   > mean(apply(matrix(rnorm(2000*6), nrow=2000), 1, var))
   [1] 1.024089
   > mean(apply(matrix(rnorm(2000*6), nrow=2000), 1, var))
   [1] 0.9980325
   > mean(apply(matrix(rexp(2000*6), nrow=2000), 1, var))
   [1] 0.9777283
   > mean(apply(matrix(rexp(2000*6), nrow=2000), 1, var))
   [1] 1.033320

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

   (a) Wind speeds at a given location were measured simultaneously on the ground and by satellite at 12 different times. The satellite transmission was not always received, however, and some values are missing (NA).

   Ground:     4.46 3.99 3.73 3.29 4.82 6.71 4.61 3.87 3.17 4.42 3.76 3.30
   Satellite:  4.08 3.94 5.00 5.20   NA 6.21 5.95   NA 4.76 3.25 4.89 4.80

   (b) A random samples of 12 heaters was selected from Brand A and another random sample of 13 heaters from Brand B. For each, the time in seconds to raise room temperature by 10 degrees was recorded.

   Brand A:  69.3 56.0 22.1 47.6 53.2 48.1 23.2 13.8 52.6	34.4 60.2 43.8
   Brand B:  28.6 25.1 26.4 34.9 29.8 28.4 38.5 30.2 30.6 31.8 41.6 21.1 36.0

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Statistics 3N03/3J04