Stat5401 Midterm Exam II - Due April 8th, 2020Exam Instruction:? There are a total number of 3 questions (25 points in total). Please checkif you have answered all the questions.? Please attach all R codes in your solution. You can use R notebook toorganize your results.? Please organize your answers in a single pdf file and submit it throughCanvas.? The exam is due at 11:59 pm on April 8, 2020 (CDT). Please try to submityour work at least a few minutes earlier than the deadline to avoid delaysdue to technical issues.? This is an open book exam. You are allowed to use your notes, book, Rhelp files, academic papers, online tutorials, etc.? You are NOT allowed to discuss the exam with anyone else.? General questions about the exam should be asked on Canvas discussionsboard ‘Midterm 2 related questions and clarifications’. Please askquestions as early as possible. I may not be able to answer last-minutequestions.? Other questions should be directed to me at lixx1766@umn.edu. If youare writing emails to me, please use your university email, i.e., ending with@umn.edu, etc. You can also send email directly through Canvas.? All answers must be written in your own words.? Updates are colored in red.1Question 1(5 points)Consider the following underlying linear regression modelyi = β1Xi1 + β2Xi2 + ?i,with the standard assumption that E(?i) = 0, V ar(?i) = σ2, and Cov(?i, ?j ) = 0for i 6= j. Note that we don’t include the intercept in this question.Suppose that you observe.Answer the following questions using math, i.e., by hand:(a) (1 points) Suppose that we fit the model:yi = β1Xi1 + β2Xi2 + ?i.Write down the design matrix X with the given X1 and X2.(b) (2 points) Fit the model yi = β1Xi1 + ?i, and let β?1 be the least squareestimator for β1.? Derive the mean and variance for β?1.? Is β?1 an unbiased estimator for β1?(c) (2 points) Suppose now that the true underlying model isyi = β1Xi1 + β2Xi3 + ?i,and you observeSuppose that you fit the modelyi = β1Xi1 + ?ito estimate β1.? Derive the mean and variance of the least square estimator for β1.? Is it an unbiased estimator?2Question 2 (Total 8 points)Download the data Q2.csv from Canvas. This is a simulated dataset motivatedby an example in the book Machine Learning with R by Brett Lantz.The response variable in the dataset is? charges = medical cost (in dollar) billed by insurance companyThe 6 covariates are? gender = Gender of the primary beneficiary, ‘f’ if female, ‘m’ if male.? age = Age of the primary beneficiary.? bmi = Body mass index? smoker = Smoker or not, ‘yes’ for smoker, ’no’ for non-smoker? children = Number of dependents, treated as a continuous/numeric covariatein this problem.? region = Residential area in the US.(a) Build a linear model by regressing charges on all the 6 covariates. Answerthe following questions.(i) Which effects are significant at α = 0.05, and what is the direction ofthe effects? Is there a relationship between age and charges?(ii) Find a 95% confidence interval for the linear coefficient for bmi.(iii) What’s the R2 and adjusted R2?(b) Build a reduced model by regressing charges on age, bmi and smoker.Compare the this model with the full model fitted in part (a) using an Ftest. According to the F test, does the model in part (a) fit the modelsignificantly better than that in part (b)?QuesStat5401作業(yè)代做纺讲、代寫R編程語言作業(yè)擂仍、data課程作業(yè)代寫、R程序作業(yè)調(diào)試 調(diào)試Matlab程序|幫做Java程tion 3 (Total 12 points)In class, we mentioned that there are many variable selection methods available.In this example, we study additional performance metrics, and use simulationto verify their effectiveness. We will also study and try the stepwise variableselection for multiple linear regression.(a) We have learned the adjusted R2 as a metric for the model fit. In thisquestion, we compare different models using the adjusted R2.Suppose the predictor is generated byset.seed(2020)n=200x=rnorm(n)3Remark: To make our results comparable, please use set.seed(2020)when generating x.(i) Let the underlying model is generated byeps=rnorm(n)y=x+x^2+x^3+epsWhat is the underlying model? How many covariates are there inthe underlying model? Please specify the covariates and true linearcoefficients.(ii) Fit 6 different models: yi = β0 +Ppj=1 βjXji + ?i for p = 1, 2, 3, 4, 5, 6.These models are polynomials of different orders. Calculate the adjustedR2for each of them, and draw a plot showing the adjustedR2. (x-axis: p, y-axis: adjusted R2) Does the correct model have thelargest adjusted R2?(Hint: You can first create a data matrixX = cbind(x,x^2,x^3,x^4,x^5,x^6)and then use a for loop to run the 6 regression models, in order tosimplify codes. Also, try summary(model)$adj.r.squared to extractthe adjusted R2for a fitted model. )(iii) Instead of using the adjusted R2, there are other performance criteria.Here, we consider the Akaike Information Criterion (AIC) andBayesian Information Criterion (BIC). Read the document at the linkhttps://daviddalpiaz.github.io/appliedstats/variable-selection-and-model-building.html.Alternatively, you can also read page 385-386 of the textbook on ‘selectingpredictor variable from a large set’ and page 705 for the defi-nition of AIC and BIC.For this question, write down the definition of AIC and BIC in termsof Residual Sum of Squares (RSS), n and p.(iv) In R, AIC and BIC can be computed using the functions AIC and BIC,respectively. Replace the adjusted R2 by AIC and BIC in part (iii)and plot the results. Does the correct model have the smallest AICand BIC?(v) Repeat the simulation in (i), (iii) and (iv) for 100 times. You will needto keep the same x while generating new random eps each time.Each time, use the adjusted R2(the largest one), AIC and BIC (smallestone) to select the model. Take record of the model selected for eachsimulation (i.e., take record of the selected p).Report the frequency that the adjusted R2, AIC, BIC correctly selectthe true model among the 100 simulations. For this problem, whichmetric selects the model best? For the other metrics, do they tend toselect more covariates or fewer covariates than the correct one?4(b) For multiple linear regression, stepwise selection (including forward search,backward search, and both directions) is usually used for model selection.Read Section 16.2 of the documenthttps://daviddalpiaz.github.io/appliedstats/variable-selection-and-model-building.html and answer the following questions(i) Use about two or three sentences to describe stepwise variable selectionmethods.(ii) Download the data Q3.csv and regress Y on X1 - X20. (Hint: trylm(Y~.,data=Q3)). Use the function step to select variables (use thedefault arguments without changing its arguments like k or direction).Report the selected variables.5轉(zhuǎn)自：http://www.6daixie.com/contents/18/5068.html