Use r/excel to simulate 1000 normal random numbers with mean 0.5 and standard deviation 0.6. record the sample mean and sample variance of these 1000 simulated random numbers. 1 (b) simulate a random number u that is uniformly distributed between 0 and 1, record its value and write it down: (c) calculate the probability that x ∼ n (0.5, 0.6 2 ) is larger than u obtained in (b). (d) use these 1000 normal random numbers obtained in (a) to find an approximate answer to (c), record its value and write it down: (e) designing a simulation procedure to more accurately approximate p(x > u). (f) find out that x is smaller than which number with probability u (note u is between 0 and 1 and hence can be treated as a probability). (g) using these 1000 normal random numbers obtained in (a) to find an approximate answer to (f), record its value and write it down: (h) given the 1000 random numbers obtained in (a), provide your 95% confidence interval of the true mean of the probability distribution that these 1000 random numbers are simulated from. (i) to reduce the width of the 95% confidence interval of the true mean in (h) by about 50%, how many more random samples from the underlying distribution are needed? (j) simulate these additional random numbers and combing them with the original 1000 random numbers to find the sample mean and sample variance, and provide your updated 95% confidence interval of the true mean.