[Solved] AMS 597 Homework 6

AMS597

1. Compute a Monte Carlo estimate of

Compare your estimate with the exact value of the integral.

2. We will estimate of

using two different approaches:

• Compute a Monte Carlo estimate () of by sampling from Uniform(0, 0.5), and estimate the variance of .
• Compute a Monte Carlo estimate () of by sampling from the exponential distribution, and estimate the variance of .
• Compare the two variances. Which one is smaller?

3. Write a function to compute a Monte Carlo estimate of the Beta(a,b) cdf, F(x)
   • by sampling from Uniform(0,x) (name this function pbeta1)
   • by sampling from U Gamma(a,1) and V Gamma(b,1) and using the result that Y = U/(U + V ) Beta(a,b) (name this function pbeta2)
   • Use pbeta1 and my.pbeta2 to estimate F(x) of Beta(3,3) for x = 0.1,0.2,,0.9. Compare the estimates with the values returned by the pbeta function in R.
4. (a) Generate X₁,,X₂₀ from N(0,1). Consider testing H₀ : = 0 vs H_a : 6= 0. Compute the p-value from (1) one sample t-test and (2) exact wilcoxon signed rank test. Repeat this process 1000 times. Estimate the empirical Type I error for both tests at = 0. (Hint: Empirical Type I error is the proportion of wrongly rejected null hypothesis).

(b) Now generate X₁,,X₂₀ from N(0.5,1). Consider testing H₀ : = 0 vs H_a : 6= 0. Compute the p-value from (1) one sample t-test and (2) exact wilcoxon signed rank test. Repeat this process 1000 times. Estimate the empirical power for both tests at = 0.05.

5. (a) Generate X₁,,X_n from N(0,1) and Y₁,,Y_n from N(0.5,1.5). Consider testing H₀ : _{X Y} = 0 vs H_a : _{X Y} 6= 0. Compute the p-value from two sample t-test. Repeat this process 1000 times. Estimate the empirical power for this test at = 0.05 for n = 10,20,30,,100. Based on your plot, what is the minimum sample size to achieve power > 80%.

(b) Edit: You do not need to show how the formula is derived. An approximate sample size formula for comparing two population means using z-test for the hypothesis in (a) is

Compare your results in (a) to this sample size formula.