A1 (a) This Monte Carlo study examines the sampling distribution of the average of 25 numbers drawn from a distribution with mean 2 and variance 9. The average of N numbers drawn randomly from a distribution with mean μ and variance σ 2 has a sampling distribution with mean μ and variance σ 2 /N. (b) Abar estimates the mean of this sampling distribution, so should be close to 2. (c) Avar estimates its variance, so should be close to 9/25 = 0.36. A3 (a) This Monte Carlo study examines the sampling distribution of the fraction of successes h in a sample of size 50 where the probability of success is 20%. When the true probability of success is p, for a sample of size N then h has a sampling distribution with mean p and variance p(1–p)/N. (b) hav should be an estimate of 0.2. (c) hvar should be an estimate of p(1–p)/N which here is (0.2)(0.8)/50 = 0.0032 (d) wav estimates the mean of the sampling distribution of the estimated variance of h, and so should be approximately 0.0032. A5 Create 44 observations from a normal distribution with mean 6 and variance 4. Calculate their average A and their median B. Repeat to obtain, say, 1000 As and 1000 Bs. Find the mean of the 1000 A values and the mean of the 1000 B values and see which is closer to 6. Calculate the variance of the 1000 A values and the variance of the 1000 B values and see which is smaller. A7 (i) Choose values for a and b, say 2 and 6 (be sure not to have zero fall between a and b because then an infinite value of 1/x2 would be possible and its distribution would not have a mean). (ii) Get the computer to generate 25 drawings (x values) from U(2,6). (If the computer can only draw from U(0,1), then multiply each of these values by 4 and add 2.) (iii) Use the data to calculate A’s estimate A* and B’s estimate B*. Save them. (iv) Repeat from (ii) 499 times, say, to get 500 A*s and 500 B*s. (v) Obtain the mean m of the distribution of 1/x2 either algebraically (the integral from a to b of 1/(b–a)x2 ) or by averaging a very large number (10,000, say) of 1/x2 values. (vi) Estimate the bias of A* as the difference between the average of the 500 A*s and m. Estimate the variance of A* as the variance of the 500 A*s. Estimate the MSE of A* as the average of the 500 values of (A* – m)2 . Compute the estimates for B* in similar fashion and compare. A9 The bias of β* is estimated as the average of the 400 β *s minus β, the variance of which is the variance of β * divided by 400. Our estimate of this is 0.01/400. The relevant t statistic is thus 0.04/(0.1/20) = 8 which exceeds the 5% critical value, so the null is rejected.