MTfall22Practice

Please do not post nor share Midterm Exam Fall, 2022 Date: October 15, at 15:00-16:30 Instructions to Students: All rights reserved. No part of this document may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without prior written permission from all contributing instructors. 1. Do not turn over cover page until instructed. 2. Ink in your name, signature, and Section in the spaces below. 3. Once the exam begins, verify that your exam copy is complete. Alert Prof if not. 4. Answer all questions on your scantron sheet, using the exam booklet to show your work. 5. Formula sheet provided (back of booklet). 6. Books and notes are not permitted. 7. No Communication Devices allowed on your person or near your desk. 8. Graphing calculators, iPads/Pods/iWatches, notebooks, computers are not allowed. Duration of examination is: 1.5 hours Statement of Academic Integrity The Telfer School of Management does not condone academic fraud. Without limiting the generality of this definition, academic fraud occurs when a student commits any of the following offences: plagiarism or cheating of any kind; use of books, notes, mathematical tables, dictionaries or other study aid unless an explicit written note to the contrary appears on the exam; has in his/her possession cameras, radios (radios with head sets), tape recorders, pagers, cell phones, or any other communication device which has not been previously authorized in writing. Statement to be signed by the student: I have read the text on academic integrity and I pledge not to have committed or attempted to commit academic fraud in this examination. Note: an examination copy or booklet without the above signed statement will not be graded and will receive a exam grade of zero. Name (LAST, first): Student Number: Section (circle): A, B, C, D, E Instructor’s name (circle): Astaraky, Brand, Khazabi, Teymouri please do not post/share individual student use only Copyright Telfer Profs

Restaurant Service Satisfaction (2 Questions) In order to improve the quality of the service, a restaurant manager regularly conducts surveys of its customers. Respondents are asked to assess food quality, service, and price. The responses are (1) Excellent, (2) Good, and (3) Fair. Surveyed customers are also asked whether they would come back. After analyzing the responses, an expert in probability determined that 87% of customers say they will return. Of those who so indicate, 57% rate the restaurant as excellent, 36% rate it as good, and the remainder rate it as fair. Of those who say they won't return, the probabilities are 14%, 32%, and 54%, respectively. 1. What proportion of customers rate the restaurant as good? a) 0.754 b) 0.654 c) 0.119 d) 0.355 e) 0. 522 2. If a customer rates the service as "Fair", what is the probability he/she will return? a) 0.221 b) 0.145 c) 0.466 d) 0.802 e) 0. 396 Theory Questions (4 Questions) 3. Which of the following best describes the concept of marginal probability? a) A de-minimis probability that can be treated as effectively zero b) It is a measure of the likelihood that a particular event will occur, if another event has already occurred. c) It is a measure of the likelihood of the simultaneous occurrence of two or more events. d) None of these choices. e) It is a measure of the likelihood that a particular event will occur, regardless of whether another event occurs. 4. If A and B are disjoint events with P(A) = 0.70, then P(B): a) can be any value between 0 and 1 b) can be any value between 0 and 0.70 c) cannot be larger than 0.30 d) cannot be determined with the information given e) Is properly obtained by way of rectangular subtraction ADM2303 Page 2 of 11 Fall, 2022 Page 2

PREDICTED RETURN: For the upcoming year, an analyst has constructed the following discrete probability distribution for firm X's predicted return. Return Probability − 5 0.20 0 0.30 5 0.40 Answer the following four questions based on the above information: 7. The expected value of this distribution is a) 1 b) 2 c) 2.5 d) 3 e) 3.5 8. The variance of this distribution is a) 14 b) 4.89 c) 17 f) 21 d) 4.58 9. The coefficient of variation for this distribution is a) 14.3 b) 7.4 c) 2.29 d) 11.5 e) 8.5 10. Assuming all returns are doubled, and probabilities remain the same what would happen to the coefficient of variation? It would be: a) doubled b) tripled c) quadrupled d) unchanged e) None above ADM2303 Page 4 of 11 Fall, 2022 Page 4

11. A recent survey shows that the probability of a college student drinking alcohol is 0.6. Further, given that the student is over 21 years old, the probability of drinking alcohol is 0.8. It is also known that 30% of the college students are over 21 years old. The probability of drinking or being over 21 years old is __________. a) 0.24 b) 0.42 c) 0.66 d) 0.90 e) 68% 12. Rumi, a production manager, is trying to improve the efficiency of his assembly line. He knows that the machine is set up correctly only 60% of the time. He also knows that if the machine is set up correctly, it will produce good parts 80% of the time, but if set up incorrectly, it will produce good parts only 20% of the time. Rumi starts the machine and produces one part before he begins the production run. He finds the first part to be good. What is the revised probability that the machine was set up correctly? a) 48% b) 56% c) 86% d) 96% e) 68% ADM2303 Page 5 of 11 Fall, 2022 Page 5

18. Which one of the following is correct? Anger score and heart disease event are most likely: a) Complementary b) Mutually exclusive c) Independent d) Dependent e) Disjoint COVID-19 SIR model A population during the pandemic can be partitioned into three groups: number of susceptible people (S), number of infected (I), and number of people who have recovered (R). Consider the population of a community in the midst of the Covid-19 pandemic where 50% of the population is considered susceptible, 20% of the population is infected, and 30% is recovered. The only rapid antigen test that is available to this community is accurate 90% of the time for susceptible individuals, 95% for infected individuals, and only 70% for recovered individuals. P(S) = 0.5 P(I) = 0.2 P(R) = 0.3 P(+ | S) = 0.9 P(+ | I) = 0.95 P(- | R) = 0.7 19. A randomly selected individual from this population is tested with the rapid antigen test. What is the probability that the result of the test is positive? a) 0.22 b) 0.45 c) 0.32 d) 0.55 e) 0.73 20. If a randomly selected individual from this population has tested positive for Covid-19, what is the probability that they are infected? a) 0.12 b) 0.34 c) 0.26 d) 0.53 e) 0.53 ADM2303 Page 7 of 11 Fall, 2022 Page 7

Air Travel During the summer, Canadian airlines across the country had to cancel hundreds of flights as they were not able to cope with the increased demand after lifting pandemic restrictions. Canadian Transportation Agency receives on average 400 calls a day to help stranded travellers deal with airlines. Suppose that the hotline is staffed for 16 hours a day. Assume that the number of calls on a time period can be modelled by Poisson distribution. (Choose the answer that is closest to your answer.) 21. What is the average number of calls in a 15-minute interval? a) 5.00 b) 1.32 c) 6.25 d) 3.67 e) 1.53 22. What is the probability of exactly 6 calls in a 15-minute interval? a) 0.74 b) 1.32 c) 0.1598 d) 0.012 e) 0.539 23. What is the probability of no calls in a 15-minute interval? a) 0.1239 b) 0.0019 c) 0.5399 d) 0.0129 e) 0.5349 24. What is the probability of at least two calls in a 15-minute interval? a) 0.123 b) 0.986 c) 0.689 d) 0.012 e) 0.033 Graduate school 40% of undergraduate students of a business school plan on pursuing a graduate degree. Fifteen undergraduate students are randomly selected from this business school. (Round your answer to 4 decimal points) 25. What is the probability that no more than two of these students plan to pursue a graduate degree? a) 0.1231 b) 0.1398 c) 0.5478 d) 0.0271 e) 0.0843 26. What is the probability that at least seven but less than nine of the students plan to pursue a graduate degree? a) 0.2311 b) 0.1381 c) 0.4781 d) 0.2951 e) 0.8491 ADM2303 Page 8 of 11 Fall, 2022 Page 8

If the price of wood is 25 dollars per square-meter and the price of pipe is 204 dollars per linear kilometer, determine the following for the total cost of materials (wood and pipe) per month (label this Z). 29. The expected value (per month) a) 21 Thousand dollars. b) 20 Thousand dollars. c) 518 Thousand dollars. d) Can not be determined. e) 188 Thousand dollars. 30. The coefficient of variation a) Can not be determined. b) 0.999 c) 1.001 d) 3.723 e) 0.269 31. Consider inflation whereby the aforementioned prices are both multiplied by 1.3. How would the coefficient of variation for monthly material (wood and pipe) costs change? It would be … a) Can not be determined. b) Increased multiplicatively by 1.3. c) Decreased multiplicatively by 0.77. d) Reduced by subtracting 1.3 from the original estimate (i.e., addition/complement rule). e) Unchanged. ADM2303 Page 10 of 11 Fall, 2022 Page 10

ADM2303 formula sheet - Fall 2022 zz Probability theory Rule of sum of probabilities: P( S ) = 1 Complement rule (Let A c be complement of A , i.e., Not A ): P( A ) = 1 − P( A c ) Addition rule for two mutually exclusive events (where ∪ connotes “or” aka union): P( A ∪ B ) = P( A ) + P( B ) Addition rule for two not mutually exclusive events: P( A ∪ B ) = P( A ) + P( B ) − P( A ∩ B ) Multiplication rule for two independent events (where ∩ connotes “and” aka intersection): P( A ∩ B ) = P( A ) × P( B ) Multiplication rule for n independent events: P( A 1 ∩ A 2 ∩ ...A n ) = P( A 1 ) × P( A 2 ) × ... × P( A n ) Multiplication rule for dependent events: P( A ∩ B ) = P( B | A )P( A ) = P( A | B )P( B ) Partition rule: for a partition B 1 , B 2 , ..., B k : P( A ) = k i =1 P( A ∩ B i ) = k i =1 P( A | B i )P( B i ) Bayes’ formula: P( B i | A ) = P( A | B i )P( B i ) P( A ) = P( A | B i )P( B i ) ∑ k i =1 P( A | B i )P( B i ) Events A and B are independent if: P( A | B ) = P( A ) and P( B | A ) = P( B ) or:P( A ∩ B ) = P( A ) × P( B ) Random variables (RV) Expected value of discrete RV X : E ( X ) = µ = n i =1 x i P( X = x i ) Variance of discrete RV X : Var(X) = σ 2 = n i =1 ( x i − µ ) 2 P( X = x i ) = n i =1 x 2 i P( X = x i ) − µ 2 Standard deviation of discrete RV X : SD ( X ) = σ = V ar ( X ) Coefficient of variation of discrete RV X : CV ( X ) = SD ( X ) E ( X ) Correlation of two discrete RV X and Y : ρ x,y = ∑ n i =1 ( x i − µ x )( y i − µ y ))P( X = x i ∩ Y = y j ) s x s y Combining random variables Adding a constant c to random variable X : E ( X ± c ) = E ( X ) ± c V ar ( X ± c ) = V ar ( X ) = σ 2 X Multiplying random variable X by a constant a : E ( aX ) = aE ( X ) V ar ( aX ) = a 2 σ 2 X Expected value of linear combination of RVs: 1 E ( aX + bY + c ) = aE ( X ) + bE ( Y ) + c Variance of linear combination of RVs (1) : V ar ( aX + bY + c ) = a 2 σ 2 X + b 2 σ 2 Y + 2 a b Cov ( X, Y ) where Cov ( X, Y ) = ρ x,y σ X σ Y . If X and Y independent then ρ x,y = 0 and covariance component drops out. Discrete and continuous distributions The Binomial probability distribution: P( X = x ) = n ! ( n − x )! x ! p x (1 − p ) n − x for x = 0 , 1 , ..., n E ( X ) = np, V ar ( X ) = np (1 − p ) The Poisson probability distribution (if approx’n of binomial, λ = np ): P( X = x ) = e − λ λ x x ! for x = 0 , 1 , 2 , ... E ( X ) = λ, V ar ( X ) = λ, e = 2 . 718 The Geometric probability distribution: P( X = x ) = (1 − p ) x − 1 p for x = 1 , 2 , ... E ( X ) = 1 p , V ar ( X ) = 1 − p p 2 The Normal distribution: X ∼ N ( µ, σ ) ⇒ f ( x ) = 1 σ √ 2 π exp − 1 2 ( x − µ σ ) 2 Z = X − µ σ ∼ N (0 , 1) ⇒ P( Z<z ) = using normal table The Exponential distribution : X ∼ Expo ( λ ) ⇒ f ( x ) = λe − λx P( X ≤ a ) = 1 − e − aλ E ( X ) = 1 λ , V ar ( X ) = ( 1 λ ) 2 The Uniform distribution: X ∼ Uniform( a, b ) ⇒ f ( x ) = 1 b − a for a ≤ x ≤ b P( x 1 <X<x 2 ) = x 2 − x 1 b − a E ( X ) = a + b 2 , V ar ( X ) = ( b − a ) 2 12 1 For cases like 2 X − 3 Y the coefficient on Y is ( − 3), thus treat accordingly; want to subtract a constant rather than add — put minus sign on c . ADM2303 Page 11 of 11 Fall, 2022 Page 11