3. (30 points) Power. In this problem, we check the power of the tests by simula

Question

3. (30 points) Power. In this problem, we check the power of the tests by simulations so as to have some practices on running simulations and to gain a better understanding of the power of a test. In practice, when we have choices of difference tests, we may want to choose the one with a higher power given that they can keep the type I error correct. In the following, we consider two scenarios (1) The sample is from the normal distribution with mean and variance 2 f(y) 2) The sample is from the Laplace distribution with mean and variance 2b 0. Suppose the We are interested in testing whether the mean is 0: H0 : sample size is 1000, so we can use the t-test under both scenarios2. The binomial test can be used for both scenarios as well since both distributions are symmetric around , so their medians are also . All tests are performed at 0.05 significance level Suppose the true distributions are (1) = 0.1, = 1 for the normal distribution; and (2) = 0.1,b-1 for the Laplace distribution, report the percentage of runs that the null hypothesis is rejected under each scenario based on 500 simulation runs. Based on the simulation results which test has a higher power under each scenario? Hint: You can use 'rnorm(1000,0.1,1)' to get a random sample from the normal distribution with = 0.1 and 1, and use 'rlaplace(1000,0.1, 1), in R package 'rmutil, to get a random sample from the Laplace distribution with 0.1 and b- = ) vs. H a :

Explanation / Answer

Level of significance is 0.05. t-table value for df=1000 and alpha = 0.05 would be t = 1.962

therefore, to fail to reject null hypothesis, for 95% of time, X_bar shall lie in interval (-t*sigma/sqrt(n), +t*sigma/sqrt(n))

1) Normal dist: sigma = 1; X_bar should lie between = (-0.062, 0.062)

2) Laplace dist: sigma = sqrt(2)*b; X_bar should lie between = (-0.088, 0.088)

By running the simulation, we get 1) for normal distribution: 439 out of 500(87.8%) simulations reject the null hypothesis 2) for laplace distribution: 294 out of 500(58.8%) simulations reject the null hypothesis.

Results of the simulations:

scenario 1) t-test power = 0.885 and binomial-test power = 0.945

scenario 2) t-test power = 0.608 and binomial-test power = 0.929

Therefore, under both the scenarios, the binomial test has the higher power.

Related Questions

Navigate

• Browse (All)
• Browse 3
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.