MAST20005/MAST90058: Assignment 3 Due date: 11am, Sunday 25 October 2020 Instructions: Please submit your assignment via the LMS, ensuring that you follow the submission instructions provided online. Remember to submit ON TIME, since late submission will receive zero points. Do not wait until the last minute! We suggest that you submit your assignment promptly once you finish all questions. You can always re-submit your assignment before the deadline if you make any changes. If for any reason you think you will not be able to submit on time, you need to notify the Subject coordinator Tingjin Chu in a timely manner (as soon as you become aware of any issue and preferably prior to the deadline). In general, a medical certificate is required. Note that extensions are only granted in exceptional circumstances and only for a very limited time period. Questions labelled with ‘(R)’ require use of R. Please provide appropriate R commands and their output, along with sufficient explanation and interpretation of the output to demonstrate your understanding. Such R output should be presented in an integrated form to- gether with your explanations. All other questions should be completed without reference to any R commands or output, except for looking up quantiles of distributions where necessary. Make sure you give enough explanation so your tutor can follow your reasoning if you happen to make a mistake. Please also try to be as succinct as possible. Each assignment will include marks for good presentation. 1 Problems: 1. (R) The daily new coronavirus cases in Victoria between Sep 7th and Sep 27th are recorded as 48 70 47 40 35 41 30 39 41 25 44 20 13 11 28 14 11 13 12 16 5 where 48 is the number of new coronavirus cases on Sep 7th, 70 is the number of new coronavirus cases on Sep 8th, and so on. (a) Use the sign test with α = 0.05 to test if the median of daily new coronavirus cases is below 15 between Sep 7th and Sep 27th. Clearly state your hypotheses. (For the purpose of this question, treat these data as a random sample from a static population, which is an unrealistic assumption in practice.) (b) Use the Wilcoxon rank-sum test with α = 0.05 to test if the median number of daily new coronavirus cases in the second week (the second row) is higher than the median number of daily new coronavirus cases in the third week (the third row). Clearly state your hypotheses. (For the purpose of this question, treat the data from each week as a random sample from a static population, which is an unrealistic assumption in practice.) 2. Let X be an exponential distribution with pdf, f(x) = λe−λx, x > 0. Suppose we observe the following random sample of n = 30: 0.11 0.21 0.75 1.14 1.35 1.63 1.63 1.83 1.93 2.04 2.16 2.25 2.41 2.52 2.65 2.83 2.92 2.92 4.83 7.23 8.80 9.80 11.54 12.16 12.91 13.93 19.68 20.94 21.73 24.09 (a) Find the p quantile, pip. (b) Calculate the ‘Type 7’ sample quantile pˆi0.25. (c) Find the asymptotic distribution of pˆi0.25. (d) Calculate a standard error for pˆi0.25. 3. Let X1, . . . , Xn be a random sample with pdf f(x) = β2xe−βx, x > 0, and for β use the prior distribution f(β) = e−β with β > 0. (a) Derive the posterior distribution of β. (b) Derive the posterior mean and the posterior standard deviation of β. 4. Consider a random sample X1, . . . , Xn ∼ N(µ, σ2). (a) If µ is unknown and σ2 is known, find a sufficient statistic for µ. (b) If µ is known and σ2 is unknown, find a sufficient statistic for σ2. (c) If µ is known and σ2 is unknown, find a sufficient statistic for σ. 2 5. (MAST20005 students only) Let X1, . . . , Xn ∼ Exp(λ) be a random sample from an exponential distribution with mean 1/λ. We are interested in testing H0 : λ = λ0 versus H1 : λ 6= λ0. (a) Derive the likelihood ratio test and show it is based on the statistic Y = ∑n i=1Xi. (b) What is the distribution of Y when H0 is true? (c) For n = 50 and λ0 = 1, find a test based on Y with significance level 0.05. 6. (MAST90058 students only) (R) Pedestrian numbers are recorded over four different locations and four different time slots over three days, as shown in the table below. Locations Time slot Flagstaff Station Melbourne Central Town Hall Bourke Street Mall 1pm – 2pm 1706 2387 3715 3715 1636 2284 3541 3689 1339 2116 3369 2884 2pm – 3pm 1063 2062 3209 2940 1065 1885 2907 2753 977 1819 3077 2525 3pm – 4pm 1380 2108 3030 2751 1306 1896 2837 2508 1261 1893 2978 2288 4pm – 5pm 2539 1980 2964 2687 2544 2025 2824 2423 2297 2064 2987 2429 Perform a two-way analysis of variance to examine whether these data suggest that pedes- trian numbers vary by time. State and test appropriate hypotheses at a 5% significance level. You should report the value of the appropriate statistic, the p-value, the assump- tions you have made and your conclusions. Is it possible to test for interaction? If yes, then perform the test and draw an interaction plot; otherwise, explain why it is not possible. 3
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