MATH 185 – Take-Home Exam 1Due Sunday, May 5th, by 11:59 PMAGREEMENTBy taking this exam, you agree to not discuss the exam with anyone, starting now,neither with a classmate or anyone else, neither in person nor through other means,including electronic. Please do not post questions on Piazza. Unless otherwise speci-fied, it is acceptable to copy-paste from the lecture or homework solution code.Problem 1. (Student vs Wilcoxon) Suppose we have a numerical sample of size n which weassume was generated iid from an underlying distribution F, unknown with a well-defined mean μ. Student’s t-test is a test about the mean: Is μ equal to a given value μ0? Wicoxon’s signed-rank test is a test for symmetry: Is F symmetric about a given μ0?That being said, the t-test can be used to test whether F is symmetric about μ0, based on thefact that ‘symmetric about μ0’ implies that ‘the mean is equal to μ0’. However, the two are notequivalent, so that the t-test is not consistent against all alternatives. Conversely, for the signedranktest to be useful as a test about the mean, we need to assume that F is symmetric about itsmean. With this additional (and nontrivial) assumption on F, testing for symmetry about μ0 isequivalent to testing whether the mean equal to μ0. (Convince yourself of that.) In what follows,we place ourselves in that situation, so that we can directly compare the two tests. There is sometheory on that. For example, it is known that when F is a normal distribution, in which case thet-test achieves the most power asymptotically (meaning in the large-sample limit), the signed-ranktest performs almost as well. We want to evaluate that with simulations.Since both tests are scale-free, we may take that F to be the normal distribution with mean μand variance 1. We consider the two-sided setting where we test μ = 0 versus μ 6= 0. For eachn ∈ {10, 20, 50, 100, 200, 500} do the following. For eacMATH 185作业代写、c/c++实验作业代做、代做Python/Java程序语言作业、代写F symmetric作业h μ in a grid of your choice, denoted M andof size 10, generate X1, . . . , Xn ～ N (μ, 1) and apply the t-test and signed-rank test, both set atlevel α = 0.10. Record whether they reject or not. Repeat this B = 1, 000 times and compute thefraction of times each test rejects. This estimates the power of each test against the alternative μ.The end result is a plot where these estimated power curves for each of these two tests are overlaid.Use colors and a legend to identify the two curves. Make sure to choose M so that we can see thepower go from about α to about 1, zooming in on the action.Note. When this problem is completed, you will have generated 6 plots all together, each withthe estimated power curves for the two tests.)Problem 2. (Fungi in brassica plants) Consider the following article about how differentbrassica plants are affected by different types of Rhizoctonia fungi.1 Read enough of the articleto understand the premise and the main findings. Otherwise, we will focus on the data given inTable 6 on how different brassica species are affected by different types of Rhizoctonia fungi.A. Write a function tableObsExp(dat) taking in a two-column data frame, with each column representinga factor, and then outputting a table of observed and expected (under no association)of counts — similar to what Table 6 in that article looks like.B. Enter the observed counts from Table 6 (likely by hand, as the data do not seem directlydownloadable) and apply your function to recover a similar table.C. Continuing with the same dataset, produce a couple of plots using functions in the ggplot2.D. Finally, ask a question and formalize it into a hypothesis testing problem. Perform a test andoffer some brief comments.1 The article was published in the scientific journal PLOS ONE and is available online at the following addresshttps://journals.plos.org/plosone/article?id=10.1371/journal.pone.0111750转自：http://www.7daixie.com/2019050556936761.html
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