Some Exercises for the Midterm 1 This is NOT a practice exam The real midterm exam will have more questions but the question types are the similar Problems in the midterm tend to be easier Some questions on here are more challenging newer topics we cover all topics on this midterm by end of Week 6 through homework and lectures 2 If you don t have time to go over all the materials start with the homework problems first and the lecture slides A complete list of review materials is available on Canvas 3 All the problems below are also important though some are unrelated to the homework All code seen is assumed to be R code 4 I have highlighted questions in red that I would consider challenge questions for this class level Do not worry if you struggle on these questions save those for last once you are comfortable with homework other questions on this review 5 A solution key for this practice is posted for you to check your understanding Part I Multiple Choice Each question only has one correct answer 1 How to generate a single random permutation of 1 2 3 A sample 1 3 size 3 replace TRUE B sample 1 3 size 3 C rbinom n 3 size 3 prob 0 5 D rpois n 3 lambda 1 2 Which distribution is discrete A Normal distribution with mean 0 and variance 1 B Normal distribution with mean 100 and variance 0 5 C Uniform distribution on the interval 0 1 D Poisson distribution with rate parameter 1 5 3 Which statement is FALSE for multinomial distribution A Multinomial is a generalization of binomial distribution to multiple categories B When we only have two categories multinomial reduces to binomial distribution C In the code rmultinom n 3 size 10 prob rep 1 4 4 the parameter size is the number of different categories D When we use dmultinom we also need to specify the prob parameter 4 Which statement is true A Sample variance is the square of sample standard deviation B Sample variance is always larger than sample mean C Sample variance is always larger than sample standard deviation D Sample variance is the square root of sample standard deviation 5 Which can be used to visually check the goodness of fit A Histogram B Rootogram C Scatterplot D None of the above 6 Suppose on average an individual has 5 mutations in the MHC region on chromosome Which is the most reasonable probabilistic model for the number of mutations in MHC for the sample of an unknown individual A Poisson with rate 5 B Poisson with rate 0 005 C Binomial with size 5 and success probability 0 1 D Binomial with size 5 and success probability 0 5 7 What error does the following code have squares c 1 4 4 5 5 5 5 9 16 squares squares 5 8 How many possible values can a random variable xtake if its distribution is binomial with n trials and probability p of success per trial 9 What is the expected number of successes of a random variable x if its distribution is binomial with n trials and probability p of success per trial A missing parenthesis B undefined variable C incorrect operator D none of the above A np B x n C n D n 1 E None of the above A np B x n C n D n 1 E None of the above 10 What is the maximum likelihood estimator of p for a random variable x if its distribution is binomial with n trials and probability p of success per trial A np B x n C n D n 1 E None of the above A B 2 C D E None of the above A B 2 C D E None of the above 11 What is the expected value mean of a random variable x if its distribution is Poisson with rate parameter 0 12 What is the variance of a random variable x if its distribution is Poisson with rate parameter 0 13 What are the possible observable values of a random variable x if its distribution is Poisson with rate parameter 0 14 What are the possible observable values of a random variable x if its distribution is binomial with size parameter s and probability parameter p A 0 1 2 B A 0 1 2 B C 1 2 3 15 How can you sample 10 times from the binomial distribution with 15 trials and probability of success 5 A rbinom n 15 size 10 prob 5 B dbinom n 15 size 10 prob 5 C rbinom n 10 size 15 prob 5 D pbinom n 15 size 10 prob 5 16 What is the maximum likelihood estimator for of a sample x1 x2 xn from a Poisson distribution with rate parameter 0 x1 2 x2 2 xn 2 A 1 n B x1 x2 xn C 1 n D None of the above x1 x2 x2 17 In ggplot2 which layering function when added to a ggplot function will plot data as a scatterplot A geom scatter B geom scatter C geom point D geom point 18 Which of the following plots is better to use for plotting discrete data A histogram B barplot 19 If Y is a binomial random variable with 30 trials and probability of success 0 45 what is the expected value of Y Show your work below A 14 5 B 30 C 13 5 D 15 E None of the above 20 If a variable named gf stores the output of the goodfit function which call would extract the MLE from the gf variable A gf var B gf par C gf D gf fitted Part II Short Answer Give necessary details to justify your answer Some of the following questions are based on concepts found in Homework 5 1 To study the genetic basis of diabetes a geneticist has collected the sequencing data at 104 SNPs SNP single nucleotide polymorphism for some paired samples A pair means a patient with diabetes and a matched in terms of age sex and other clinical covariates healthy individual Assume these SNPs are mutually independent We perform a statistical test at each SNP using the null hypothesis that the SNP is not associated to diabetes and record the p value So in total we have 10 4 p values Suppose the smallest 5 p values ordered we have collected are given by 8 107 2 10 6 2 105 4 105 9 10 5At significance level 0 05 which of the above p values are significant according to Bonferroni correction And when do we use the Bonferroni Correction 2 For some genetic locus there are two possible alleles A and a The frequency of A in the entire population is 0 6 and the frequency of a is 0 4 If we have sampled 30 unrelated individuals assuming Hardy Weinberg equilibrium what is the expected frequency of the genotype Aa in this sample Show your work 3 Define what a type I and type II error …