STAT 230 Spring 2023 Quiz 4 Answers (A)

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University of Waterloo *

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Statistics

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Jan 9, 2024

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Instructions Marking Scheme: 1. All electronic devices are to be turned off and put away. 2. This is a closed-book quiz. 3. Answer the questions in the spaces provided. 4. When considering only a single random variable, ࢄ , ࢌ ࢄ ሺ࢞ሻ ൌ ࢌሺ࢞ሻ and ࡲ ࢄ ሺ࢞ሻ ൌ ࡲሺ࢞ሻ . 5. Final answers to written questions should be rounded to THREE DECIMAL PLACES or left in fraction form. 6. Only question pages will be marked. 7. You may tear off the last page and use it for rough work. All pages will be collected at the end of the quiz. 8. Only a non-programmable, non-graphical calculator with a pink-tie or blue-goggle sticker will be allowed. No other aids are allowed. 9. A formula sheet has been provided at the end of this quiz. 10. DO NOT WRITE ON THIS COVER PAGE. Questions Out of 1 9 9 10 10 11 7 Total 26 CROWDMARK Please print in pen: Waterloo Student ID Number: WatIAM/Quest Login Userid: Examination Quiz 4 (Version A) Spring 2023 STAT 230 Times: Friday 2023-06-30 at 10:30 to 11:20 Duration: 50 minutes Exam ID: 5398050 Sections: STAT 230 LEC 001-003 Instructors: Erik Hintz, James Adcock, Jeremy VanderDoes Special Materials Candidates may bring only the listed aids. Calculator - Pink Tie (or other Faculty approved calculators) · STAT 230 Spring 2023 Quiz 4 © 2023 University of Waterloo Please initial: Page 1 of 12

Part 1: Multiple Choice Each question is worth 1 mark Please CLEARLY CIRCLE your answer selection on this page. Do NOT simply write your choice next to the question. If you do, the question will not be graded. Each question has one correct answer. Choose the best answer. 1. Car accidents at a certain intersection are randomly distributed in time according to a Poisson process, with an average of 5 accidents per week. What is the probability of exactly 9 accidents in a 3-week period? A) 0.0324 B) 0.0246 C) 0.8162 D) 0.4243 2. In your first course at the University of Waterloo, the Professor bases the course grade on one quiz, one midterm, and one final exam. The quiz counts for 25%, the midterm counts for 30%, and the final exam counts for 45% of your course grade. Your quiz score was 87% and your midterm score was 62%. If the final exam is out of 100 marks, what is the minimum grade you need to score on the final exam to get 80% in the course? A) 86% B) 89% C) 75% D) 80% 3. Cans of soda sell for $1.50 each in a vending machine. The operator pays $0.40 for each can of soda and $10 a day to operate the machine. The expected number of cans sold each day is 150. What is the expected daily profit of the machine? A) $155 B) $165 C) $215 D) $175 4. A coin in flipped ݊ times. In ݊ independent flips on the coin, the expected number of heads is 6.8, with a variance of 2.5. If we let ݌ ൌ ܲሺ heads is flipped ሻ , what is ݌ ? A) ͲǤ͵͸ͺ B) ͲǤ͸͵ʹ C) ͲǤͻͳͻ D) Cannot be determined from the given information CROWDMARK STAT 230 Spring 2023 Quiz 4 © 2023 University of Waterloo Please initial: Page 3 of 12

5. Phil has challenged James to a cribbage contest. Note : Cribbage is a popular card game. The rules of the game are not important for this question. Before the start of the contest, they each put $10 into the pot. They have decided that the first person to win 10 games wins the pot. It is assumed that they are equally matched so that the probability that either wins any game is 1/2. There are no ties. After playing 16 grueling, hard-fought games, they are forced to quit due to fatigue, with James having won 7 games and Phil having won 9 games. They decide to split the pot fairly and go home for the night. W ± ǯ f the total $20 pot be? A) $8.75 B) $4.44 C) $2.50 D) Cannot be determined from the given information 6. A secretary types words at an average rate of 60 words per minute. They make a mistake on any particular word with probability 0.04, independent from word to word. Thankfully, the secretary has a very keen eye and can detect all mistakes that are made while typing a document. If a mistake is made, it must be corrected. It takes 15 seconds to correct each mistake. Let ܺ represent the time (in minutes) to finish a 600-word document. What is ܸܽݎሺܺሻ ? A) ͳͳǤͶͶ B) 4.61 C) 345.60 D) ͳǤͶͶ 7. A boat owner wishes to purchase an insurance policy for their boat for the upcoming year. The policy requires an annual premium to be paid to the insurance company. The policy will cover only one of the following four events in any year: total loss, large damage, small damage, and no damage. If there is a total loss event, the replacement cost of $8,000 will be paid by the company. If it is not a total loss, but the damage exceeds $3,000, then $2,100 will be paid by the company. Nothing will be paid for damage less than or equal to $3,000. And of course, nothing is paid for no damage. The company estimates the probability of the above events to be 0.03, 0.21, 0.35, and 0.41, respectively. What should the insurance company set the annual premium to be if they wish to make a profit of $50 per policy, on average? A) $239 B) $544 C) $731 D) $920 CROWDMARK STAT 230 Spring 2023 Quiz 4 © 2023 University of Waterloo Please initial: Page 4 of 12

8. Suppose that the number of surface flaws ǯ Poisson distribution with a mean of 0.06 Ǥ ² ǯ contains 10 square feet of plastic panels. If ܺ denotes the number of cars that need to be examined in order to find one with no flaws, what is ܧሺܺሻ ? A) 1.822 B) 0.942 C) 1.062 D) 0.822 9. Suppose the random variable X has probability function of the form: ݂ሺݔሻ ൌ ݇ ቀ ܽ ݔ ቁ , for ݔ ൌ Ͳǡͳǡʹǡ ǥ ǡ ܽ ; ݂ሺݔሻ ൌ Ͳ otherwise. Note that ܽ and ݇ are parameters in the above probability function. What is ܧ൫ܺሺܺ െ ͳሻ൯ in terms of ܽ ? A) ௔ሺ௔ିଵሻ ଶ B) ௔ ଶ ቀͳ ൅ ௔ ଶ ቁ C) ௔ ଶ D) ௔ሺ௔ିଵሻ ସ CROWDMARK STAT 230 Spring 2023 Quiz 4 © 2023 University of Waterloo Please initial: Page 5 of 12

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