STA238 – Probability, Statistics and Data Analysis II Winter Midterm March 16, 2022 10:10 AM – 11:40 AM ET, Submit by 12:00 PM ET This question document contains 7 pages (including this cover page) and 5 problems. The midterm is an individual and independent open-book test. The expectation is the problems below are completed by hand with your work shown (e.g. integration steps, etc.). Your work must be submitted on Crowdmark no later than 12 PM ET. This is a strict deadline. The term ‘open-book’ means you have access to the following resources within our course ONLY: Course notes Course Textbooks Probability calculations in R (not mandatory), scientific calculator, probability tables (found on our Midterm Information Page) Using any other resources beyond our course is considered an unauthorized aid. This includes but is not limited to: Posting on discussion board or elsewhere for help (e.g., Slack, Discord, Facebook, etc.) Working with a classmate (this is an individual and independent assessment!). The work you submit must be entirely your own Receiving assistance, tips, hints, or otherwise from someone else Submitting any work that is not your own Searching for answers on other platforms (e.g. external textbooks, TAs, Google, etc.) u Clearly state and define any variables and/or distributions used, including pa- rameters . Use appropriate probability and event notation in your work. u Show all your work, including calcula- tions, integrations, etc. We can only assess when you can show us u Round all final answers to 4 decimal places (or 2 decimal places if expressed as a percentage). u Send any technical issues or urgent ques- tions that occur during the test to the course email (sta238@utoronto.ca)! Problem Points Score 1 11 2 3 3 8 4 8 5 9 Total: 39 1. (11 points) You examine historical data and find that the wait time for a common service has a Gamma distribution with = 1.5 and = 2. The graph of the density function for a Gamma( = 2, = 2) is plotted below, with the red vertical line indicating the mean of the density function. Note: If X Gamma( , ), E(X) = , V (X) = 2 (a) (3 points) TRUE/FALSE and explain your choice: According to Central Limit Theorem, X30 will have an approximately normal distribution. (b) (5 points) What is the probability that you will observe a sample mean of 30 observations that is within 0.5 of the true mean Justify your calculation and explicitly state the distribution used. STA238 – Probability, Statistics and Data Analysis II Page 2 of 7 True ni is sufficient enough hyc.LT if the single size is big enough then thesampling distrim ofthevariable’s man will approximate h normal distribution regardless of variable’s dim hthpqubllun oxi.hu M P 以 苦 0.2 P 1I 1 Mail P 25 i 35 的 standarte 王 器 王 2器 王 1.12 亚1.12三器磊 314 (c) (3 points) What information about the interval in (b) does Chebyshev’s inequality give you Calculate it. 2. (3 points) The function is designed to return what output Write your selection here: Briefly explain your choice. a) Density histogram of sample variance for a dataset of size n b) Density histogram of simulated data from an unknown distribution c) Density histogram of simulated data from a known distribution d) Simulated data points from some distribution STA238 – Probability, Statistics and Data Analysis II Page 3 of 7 Pl It17 的 岳 资 0.8 Pll In 31 fcikl llxiulio.ir 0.2 a 0 Because we knw the formula and we calculate th Sank variable foreach sampler and save them and repeat he process drew the histogram 3. (8 points) It is important that face masks used by firefighters be able to withstand high tem- peratures because they commonly work in temperatures of 90 to 250 degrees Celsius. In a test of one type of mask, 11 of the 55 masks had the lenses pop out at 120 degrees Celsius. (a) (5 points) Construct a 92% confidence interval for the true proportion of masks of this type that will fail at 120 degrees Celsius in this way. Use the method that results in the most conservative interval. (b) (3 points) Do you think the interval you’ve constructed has exactly 92% confidence If not, do you think it is higher or lower Explain. Note: Your response should not rely on any calculations, but instead provide a rational justification for your response. STA238 – Probability, Statistics and Data Analysis II Page 4 of 7 i 前 阳 ni 卡上175檿 CICU.IO510294 仙圃 的 pushy or to be more ansewdhe It is not exactly 92 it is under bean In orderto bemore consent we use o.it i so th error will increase 4. (8 points) Below is a data set on testosterone measurements on 22 male athletes, sorted in numerical order. Use this data to construct a fully labeled modified boxplot representing the distribution of testosterone among these 22 male athletes. Use the percentile calculation method for any percentiles you need to compute. How would you describe the shape of the distribution Note: The [#] at the beginning of each line indicates the element number starting in that row. For example, 190.0 is the 11th sorted data point. STA238 – Probability, Statistics and Data Analysis II Page 5 of 7 5 6 mean 206.418 minimum 112 maximum 487 median 190 191.6 Q Pxlntl 0.25 1 5.75 kite 8 Xan 1 137.6 G pxlntl o75 22 1 1725 Xaneilxcn.hn1 271.98 10次 G G 13.435 011 1.5 63.925 以 欣 473.475 No aelieriii.lu ai 5. (9 points) Let X1, X2, …, X8 be a random sample from a distribution with parameter . The density function is given by: f(x ) = ~ 3 2 x3e x, x > 0 0, otherwise Note E(X) = 3 and V (X) = 3 2 (a) (2 points) Write out the likelihood and log-likelihood function of . (b) (5 points) Derive the maximum likelihood estimator for . Find also the maximum likeli- hood estimator for the variance of X. What property allows you to do this easily STA238 – Probability, Statistics and Data Analysis II Page 6 of 7 点 州6 点些 in637点灯 了 e 0 1g 1 3ngonly您们 hga_噝 讻 点 Y 0 E 器 年 (c) (2 points) Your random sample yielded the following observations: 0.78 0.60 0.89 0.63 0.28 0.65 0.36 1.16 Find the maximum likelihood point estimate for and for E(X). STA238 – Probability, Statistics and Data Analysis II Page 7 of 7 y uotubtehhtotfy oh5 品顺 点 章 0.66875
