1. (a) Define the following terms: homoscedasticity, level of significance, P-value, power. [4 marks]

   (b) Describe an application where the pivotal quantity (s₁²/s₂²)/q ~ F(n₁-1, n₂-1) could be used. [3 marks]

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

   > var(apply(matrix(rnorm(2000*6), nrow=2000), 1, mean))
   [1] 0.1704955
   > var(apply(matrix(rnorm(2000*6), nrow=2000), 1, mean))
   [1] 0.1628808
   > var(apply(matrix(rexp(2000*6), nrow=2000), 1, mean))
   [1] 0.1618470
   > var(apply(matrix(rexp(2000*6), nrow=2000), 1, mean))
   [1] 0.1694257

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) A press used to remove water from copper-bearing materials is being tested using two types of filter plates. Twenty-two samples of material were randomly assigned, 11 to the regular chamber and 11 to the diaphragm chamber. The data show the percentage of moisture remaining in the material after treatment..

   Regular chamber:   8.10 7.96 7.97 8.02 7.82 8.15 8.16 7.98 8.08 7.87 8.11
   Diaphragm chamber: 7.58 7.66 7.58 7.65 7.63 7.46 7.65 7.67 7.62 7.58 7.54

   (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