Exam 1 Fall 2023 Solution

STAT 350 Exam 1 V1 Fall 2023 Name: ___________________________________ PUID ____________________________ Instructor (circle one): Anand Dixit Timothy Reese Halin Shin Khurshid Alam Class Start Time: ○ 11:30 AM ○ 12:30 PM ○ 1:30 PM ○ 2:30 PM ○ 3:30 PM ○ Online As a boilermaker pursuing academic excellence, I pledge to be honest and true in all that I do. Accountable together - we are Purdue. Instructions: 1. IMPORTANT Please write your name and PUID clearly on every odd page. 2. Write your work in the box. Do not run over into the next question space. 3. You are expected to uphold the honor code of Purdue University. It is your responsibility to keep your work covered at all times. Anyone caught cheating on the exam will automatically fail the course and will be reported to the Office of the Dean of Students. 4. It is strictly prohibited to smuggle this exam outside. Your exam will be returned to you on Gradescope after it is graded. 5. The only materials that you are allowed during the exam are your scientific calculator , writing utensils, erasers, your crib sheet , and your picture ID . If you bring any other papers into the exam, you will get a zero on the exam. Colored scratch paper will be provided if you need more room for your answers. Please write your name at the top of that paper also. 6. The crib sheet can be a handwritten or type double-sided 8.5in x 11in sheet. 7. Keep your bag closed and cellphone stored away securely at all times during the exam. 8. If you share your calculator or have a cell phone at your desk, you will get a zero on the exam. 9. The exam is only 60 minutes long so there will be no breaks (including bathroom breaks) during the exam. If you leave the exam room, you must turn in your exam, and you will not be allowed to come back. 10. You must show ALL your work to obtain full credit. An answer without showing any work may result in zero credit. If your work is not readable, it will be marked wrong. Remember that work has to be shown for all numbers that are not provided in the problem or no credit will be given for them. All explanations must be in complete English sentences to receive full credit. 11. All numeric answers should have four decimal places unless stated otherwise. 12. After you complete the exam, please turn in your exam as well as your table and any scrap paper that you used. Please be prepared to show your Purdue picture ID . You will need to sign a sheet indicating that you have turned in your exam. Your exam is not valid without your signature below. This means that it won't be graded. I attest here that I have read and followed the instructions above honestly while taking this exam and that the work submitted is my own, produced without assistance from books, other people (including other students in this class), notes other than my own crib sheet(s), or other aids. In addition, I agree that if I tell any other student in this class anything about the exam BEFORE they take it, I (and the student that I communicate the information to) will fail the course and be reported to the Office of the Dean of Students for Academic Dishonesty. Signature of Student: _________________________________________________________

Version: V1 2 You may use this page as scratch. The following is for your benefit only; we will not use this for grading: Question Number Total Possible Your points Problem 1 (True/False) (2 points each) 12 Problem 2 (Multiple Choice) (3 points each) 12 Problem 3 18 Problem 4 16 Problem 5 16 Problem 6 26 Total 100

Version: V1 4 2. (12 points, 3 points each) Multiple Choice Questions. Please indicate the correct answer by filling in the circle. If you indicate the correct answer by any other way, you may receive 0 points for the question . For each question, there is only one correct option given . 2.1. Suppose X is a continuous random variable with mean μ = 5 and standard deviation σ = 5 . What is the value of P(X = 10) ? Ⓐ P(X = 10) = 0.68 Ⓑ P(X = 10) = 0.8413 Ⓒ P(X = 10) = 0.95 Ⓓ P(X = 10) = 0 Ⓔ Not enough information to calculate it. 2.2. Suppose ? ? , ? ? , … , ? 𝒏 is a random independent sample coming from the same (identically distributed) but unknown distribution with finite non-zero variance σ 2 . Identify the incorrect statement . Ⓐ For any n , ?[? ̅ ] = ?[? ? ] . Ⓑ 𝐕𝐚?(? ? − ? ? ) ≠ ? Ⓒ For any n , X ̄ will be approximately normally distributed. Ⓓ As n increases the random variable ( X ̄ - μ ) / σ becomes approximately normally distributed with mean 0 and standard deviation 1. Ⓔ For any n , 𝐕𝐚?(? ̅ ) ≤ 𝐕𝐚?(? ? ) . 2.3. The measure of spread which is the most likely to be influenced by outliers in the dataset is Ⓐ sample mean Ⓑ sample median Ⓒ sample standard deviation Ⓓ Inner quartile range (IQR) Ⓔ more than one of the above. 2.4. Let X and Y be two normal random variables with means μ X and μ Y , and standard deviations σ X and σ Y , respectively. Select the correct statement regarding the relationship between the parameters ( μ X , μ Y , σ X , σ Y ) when the normal curve associated with X is more peaked than the normal curve associated with Y . Ⓐ σ X < σ Y Ⓑ σ X > σ Y Ⓒ ( μ X / σ X ) > ( μ Y / σ Y ) Ⓓ μ X < μ Y Ⓔ μ X > μ Y

Version: V1 Name: _____________________________ PUID : _________________ 5 3. (18 points) Halin goes fishing along the river in her area every weekend. Once she begins fishing, it is known that she needs to wait 20 minutes on average to catch a fish. a) (2 pts) Define the random variable X that represents the number of fish Halin catches in two hours of fishing. Write the name of its distribution and provide the value of the parameter, λ . There is an average of 1 fish caught every 20 minutes so there is an average of 3 fish caught per one hour therefore ? = ? ?? × ??? = ? . ? ∼ 𝐏?𝐢????(? = ?) b) (4 pts) What is the probability that Halin catches exactly 8 fish in two hours? The pmf for a Poisson is ?(? = 𝒙) = ? 𝒙 ? −? 𝒙! ?(? = ?) = ? ? ? −? ?! = ?. ???? c) (8 pts) While talking with her friends, Halin recalls that on a lucky weekend, she had caught somewhere between 7 and 9 fish (inclusive) in two hours. Given this information, what is the probability that Halin actually caught 8 fish that weekend? ?(? = ?|? ≤ ? ≤ ?) = ?({? = ?}⋂{? ≤ ? ≤ ?}) ?(? ≤ ? ≤ ?) = ?(? = ?) ?(? = ?) + ?(? = ?) + ?(? = ?) = ?. ???? ?. ???? + ?. ???? + ?. ???? = ?. ???? ?(? = ?) = ? ? ? −? ?! = ?. ???? ?(? = ?) = ? ? ? −? ?! = ?. ???? d) (4 pts) It is known that 35% of the fish caught in this river are largemouth bass. Halin now remembers that she did catch exactly 8 on that lucky weekend. What is the probability that 3 of the eight she caught were largemouth bass? Hint: Each fish she caught was either a largemouth bass or it was not. Notice that she knows 8 fish were caught but we do not know how many were largemouth bass. Each fish is either largemouth bass or some other fish ( B inary), each catch can be considered I ndependent of other catches, and there is a fixed sample size 𝒏 = 8 , and the probability of a fish being largemouth bass is 𝑝 = 0.35 ( S uccess). The situation is BInS (Binomial experiment). Let 𝑌 denote the number of largemouth bass caught ? ∼ ?𝒊𝒏(𝒏 = ?, 𝒑 = ?. ??) . ?(? = ?) = ( ? ? ) ?. ?? ? ?. ?? ?−? = ?? × ?. ?????? × ?. ????? = ?. ????

Version: V1 Name: _____________________________ PUID : _________________ 7 5. (16 points) Suppose a college student uses the bus 70% of the time to go to his first class of the week on the college campus. Further suppose the student chooses to walk whenever he chooses not to take the bus. Assume that the student can choose only one form of transportation at a time. If the student chooses to walk, he is late 30% of the time. Further, if he chooses to use the bus, he is late 20% of the time. Using this information, answer the following questions. a) (6 pts) What is the probability that the student will be late when he travels to his first class of the week on the college campus? Let 𝐿 = the student is late, 𝐵 = the student takes the bus, and 𝑊 = the student walks. Given: ?(𝐵) = 0.7 , ?(𝑊) = 0.3 ?(𝐿|𝐵) = 0.2 , ?(𝐿|𝑊) = 0.3 Compute: Use the law of total probability ?(𝑳) = ?(𝑳|?)?(?) + ?(𝑳|?)?(?) = ?. ? × ?. ? + ?. ? × ?. ? = ?. ?? b) (4 pts) What is the probability that the student will not be late when he travels to his first class of the week on the college campus? Complement Rule: ?(𝐿 ′ ) = 1 − ?(𝐿) = 0.77. c) (6 pts) What is the probability that the student chooses to walk, given that he was found to be on-time for his first class of the week on the college campus? Compute: Bayes Rule ?(?|𝑳 ′ ) = ?(𝑳′|?) ?(?) ?(𝑳 ′ ) = ?. ? × ?. ? ?.?? = ?. ???? Where ?(𝐿′|𝑊) = 1 − ?(𝐿|𝑊) = 1 − 0.3 = 0.7

Version: V1 8 6. (26 points) At a beverage company, a vigilant quality control engineer has detected irregularities in the bottle filling process. The production line includes two distinct zones. In Zone A, bottles consistently leave the machine underfilled, with a deviation of 1 to 3 ml below the desired level. Conversely, in Zone B, bottles consistently exit overfilled, exceeding the target by 1 to 3 ml. To understand these variations, we utilize a functional relationship, denoted as ? ? (𝒙) that characterizes the probabilities associated with the fill levels of these bottles. However, the normalizing constant must be established before any probability calculations can be performed. ? ? (𝒙) = { 𝒌 −? ≤ 𝒙 ≤ −? 𝒌 (−??𝒙 + ??) ? ≤ 𝒙 ≤ ? ? otherwise a) (4 pts) Determine the value of 𝒌 such that ? ? (𝒙) is a valid probability density function. Clearly 𝑘 > 0 sketch the function out if you cannot see this. Now solve ∫ 𝑘𝑑𝑥 −1 −3 + ∫ 𝑘(−15𝑥 + 45)𝑑𝑥 3 1 = 1 2𝑘 + 15𝑘 [− 𝑥 2 2 + 3𝑥] 1 3 = 1 3𝑘 + 30𝑘 = 1 𝒌 = ? ?? ? ? (𝒙) = { [?] 𝒙 ≤ −? [?] −? ≤ 𝒙 < −? [?] −? ≤ 𝒙 < ? − ?? ?? 𝒙 ? + ?? ?? 𝒙 − ?? ?? ? ≤ 𝒙 < ? [?] 𝒙 ≥ [?] b) (6 pts) Determine the missing parts [A, B, C, D, E] of the cumulative distribution function. [?] = ? [?] = ? [?] = ? [?] = ∫ ( ? ?? ) ?𝒕 = ? ?? (𝒙 + ?) 𝒙 −? [?] = ∫ ( ? ?? ) ?𝒕 = ? ?? (−? + ?) = ? ?? −? −? (CDF is constant in the region −? ≤ 𝒙 < ? )

Version: V1 10 Zone A → 5-number summary approximately [𝐌𝐢? = −?. ??, ? ? = −?. ??, ? ? = −?. ?, ? ? = ?. ??, 𝐌𝐚𝐱 = ?. ?] Zone B → 5-number summary approximately [𝐌𝐢? = ?. ??, ? ? = ?. ??, ? ? = ?. ??, ? ? = ?. ??, 𝐌𝐚𝐱 = ?. ?] Zone A looks almost symmetric around 0 with a slight positive skew because median is closer to ? 1 . There aren’t explicit points so there aren’t any real outliers. Zone B is right skewed with four high real outliers. The large gap between these points and the whisker indicates that they are real outliers. Zone A seems to have been fixed as it is roughly centered around 0. Zone B still seems to have issues as it is now right skewed and is still overfilling the bottles.