Assignment2_Winter2024 V3

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Carleton University *

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151

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Statistics

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Feb 20, 2024

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STAT 151 STAT 151 Assignment 2 Due date : refer to the Course Outline Purposes This assignment has two parts. The following questions assess your ability to identify the sample space of a chance experiment, calculate probabilities using the equally likely outcome model, and the addition, complementation, conditional probability, and multiplication rules, determine if two events are independent by calculation, and apply counting rules. This assignment also assesses your understanding of discrete probability distributions, your ability to find the mean (expected value) and the standard deviation of a discrete random variable and your ability to identify and work with binomial random variables. The second part assesses your ability to use R commander to compute the probabilities listed above. Instructions For every assignment in this course, you are required to complete the questions or tasks in Part A by hand. This means that to do any calculation or drawing, you will NOT use R commander or any computer application. That is, you are meant to do the calculations manually with a non- programmable scientific calculator and use a pen or pencil to draw figures or build a distribution table on paper (or on an iPad/tablet). Then you will submit a photo of your written solution using the appropriate submission box on the corresponding Crowdmark submission page. Before you complete Part B using R commander, you should read and practice the R commander steps by following the related examples in the Lab Manual and the Demos, which you can download via a link in the Course Content folder on mêskanâs. Where appropriate, units should be included in your answer. Show your calculations fully and provide a concluding sentence to your problems. Part A 1. The handedness of a group of 3 people are Ambidextrous (A), Lefthanded (L), or Righthanded (R). Suppose two people are randomly selected with replacement (that is, a person is selected, their handedness is observed and they are returned to the group of three people). Let A be the event of selecting an ambidextrous person, L be the event of selecting a lefthanded person, and R be the event of selecting a righthanded person. (a) List all possible outcomes. (2 marks) (b) List all possible outcomes for each of the following events and find the corresponding probabilities. (8 marks) 1

STAT 151 i. E1 = {Exactly 1 person of type L is drawn} ii. E2 = {The second person is right handed} iii. E3 = {At MOST one person is ambidextrous} iv. E4 = {Both people have the same handedness} (c) List all possible outcomes and find the probabilities of the following events. (12 marks: 2 for each) i. Not E3 ii. E1 & E3 iii. E1 & E4 iv. E2 & E3 v. E2 & E4 vi. E3 or E4 (d) Identify all possible pairs of events defined in part (b) that are mutually exclusive. (3 marks) (e) Verify mathematically that E2 and E4 are independent events, while E3 and E4 are not. (3 marks) 2. Suppose we select a random sample of 4 numbers between 1 and 30, sampling without replacement. (a) How many unordered samples of size 4 are possible? (2 marks) (b) What is the probability that the sum of the values in our sample is less than 14? (3 marks) 3. People can be right handed, mixed handed, or left handed: RH, MH, LH. People can also be right footed, mixed footed or left footed: RF, MF, LH. So a person can be one of 9 handedfooted possibilities: RHRF, RHMF, RHLF, MHRF, MHMF, MHLF, LHRF, LHMF, LHRF. Among a group of 1000 students you find that 600 are RHRF, 265 are RHMF, 31 are RHLF, 3 are MHRF, 17 are MHMF, 4 of them are MHLF, 14 of them are LHRF , 19 of them are LHMF, and 47 of them are LHLF. A random sample of 9 students will be selected from this group of 1000 students. Later in the course we will see how these figures compare to the figures found in Table 3 in the 2016 research paper “Footedness Is Associated with Self-reported Sporting Performance and Motor Abilities in the General Population”, by Ulrich S. Tran and Martin Voracek https://www.frontiersin.org/articles/10.3389/fpsyg.2016.01199/full , but for now we will limit ourselves to questions regarding sampling choices. 2

STAT 151 (a) How many different samples are possible? (2 marks) (b) How many different samples of size 9 are possible subject to the constraint that no 2 students may have the same handedfooted result? (2 marks) (c) What is the probability that a random sample of 9 students from this group has no two students with the same handedfooted result? (2 marks) 4. A certain lottery sells 10 million tickets for $2 each. Let X denote your winnings upon purchasing 1 ticket, and suppose X has the following probability distribution x P ( X = x ) $ 2,000,000 1 10,000,000 $ 100,000 10 10,000,000 $ 1,000 20 10,000,000 $ 0 9,999,969 10,000,000 (a) What is the expected value of X. (2 marks) (b) Since each ticket costs $2, define Y=X-2 to be your profit from purchasing 1 ticket. Compute and interpret the expected value of Y. (2 marks) (c) What is the probability that you have no winnings if you purchase a ticket? (So you win $0 on your ticket) (1 mark) (d) Suppose you purchase 30 tickets. Let T be the total winnings among all 30 tickets. What is the probability that you have 30 losses? (So no ticket wins any money). For simplicity, assume independence. (2 marks) (e) What is the probability you win any money among all the 30 tickets (so at least one ticket wins an amount). (3 marks) 5. At 11 pm one evening, the Queen discovers that her carrying case of bespoke gloves has gone missing after her recent return from a trip overseas. She is due to leave on a 30-day trip to Korea at 5 pm the next night. Her dresser Angela contacts her glove maker 3

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STAT 151 Genevieve, who says that she can order an emergency shipment of white sueded cotton from the United States to be flown overnight, and that her best emergency seamstresses, Catherine, Beatrice, and Eugenie, can sew new gloves tomorrow and that they will make a delivery of the gloves they make to the Queen at 4 pm tomorrow. If the shipment arrives on the 8 am train (best scenario), Catherine, the head seamstress, can make 24 pairs, Beatrice, the next most experienced seamstress, can make 20 pairs, and Eugenie, the next most experienced seamstress, can make 16 pairs before 4 pm. If the shipment arrives on the 10 am train (medium scenario), Catherine can make 18 pairs, Beatrice can make 15 pairs, and Eugenie can make 12 pairs before 4 pm. If the shipment arrives on the noon train (worse scenario) Catherine, the head seamstress, can make 12 pairs, Beatrice can make 10 pairs, and Eugenie can make 8 pairs before 4 pm. There is a 40% chance the shipment of white sueded cotton will arrive by 8 am, a 30% chance the shipment will not arrive until 10 am, and a 30% chance the shipment will not arrive until noon. Catherine Beatrice Eugenie Best Case 24 20 16 Medium Case 18 15 12 Worse Case 12 10 8 After a lifetime of service, Her Majesty Queen Elizabeth II sadly passed away at the age of 96 on September 8, 2022. This question is inspired by the article cited below. https://www.rd.com/article/queen-elizabeth-gloves/ a) Determine the expected number of gloves that can be made by each of the 3 seamstresses tomorrow. (6 marks) b) Will the Queen have enough gloves for 30 days, if she needs a new pair every day? (2 marks) c) Compute the standard deviation for Eugenie’s gloves. (3 marks) 6. Linus Ullmark of the Boston Bruins had the best save percentage of a goalie in the 2022-2023 season. We take this to mean, loosely, that Ullmark has a 93.8% chance of making a save when a puck is shot at his goal. In a practice in the 2023-2024 season, a fellow Bruin makes 20 shots on Ullmark. a) What is the probability that Ullmark will make saves on more than 18 shots on goal? (4 marks) b) What is the probability that he will miss stopping (allow) at least 2 of the shots on goal? (4 marks) c) How many shots on goal do you expect Ullmark to stop? (2 marks) 4

STAT 151 d) Obtain the standard deviation of the number of shots on goal that Ullmark will stop. (2 marks) Part B Complete the following questions using R and R commander. 1. People can be right handed, mixed handed, or left handed: RH, MH, LH. People can also be right footed, mixed footed or left footed: RF, MF, LH. So a person can be one of 9 handedfooted possibilities: RHRF, RHMF, RHLF, MHRF, MHMF, MHLF, LHRF, LHMF, LHLF. Table 3 in the 2016 research paper “Footedness Is Associated with Self-reported Sporting Performance and Motor Abilities in the General Population”, by Ulrich S. Tran and Martin Voracek summarizes a study of handedness and footedness in 12720 people. https://www.frontiersin.org/articles/10.3389/fpsyg.2016.01199/full . The dataset TRANVORACEKHANDFOOT contains raw data from which the following two way table of counts for the 9 cells RHRF, RHMF, RHLF, MHRF, MHMF, MHLF, LHRF, LHMF, LHRF can be obtained. Your instructor added the totals. LF MF RF TOTALS LH 602` 247 177 1026 MH 45 219 33 297 RH 396 3373 7628 11397 TOTALS 1043 3839 7838 12720 (a) Check that you can get the results of the table of Tran and Voracek by generating a two- way table in R from the TRANVORACEKHANDFOOT dataset. Paste your result. (3 marks) (b) Find, by hand, the probability that a randomly selected person is left handed. (2 marks) (c) Find, by hand, the probability that a randomly selected person is left handed and right footed. (2 marks) (d) Find, by hand, the probability that a randomly selected person is left handed or right footed. (3 marks) (e) Find, by hand, the probability that a randomly selected left-handed person is mixed footed. That is, find the probability that a randomly selected person is mixed-footed given they are left-handed. (3 marks) (f) Find, by hand, the probability that a randomly selected mixed-footed person is left- handed. (3 marks) (g) Are events MF and LH independent? Explain your answer by hand. (2 marks) 5

STAT 151 (h) Are events LH and M F mutually exclusive? Explain your answer by hand. (2 marks) 2.The data set HEARTFAILUREPREDICTION looks at several variables that play a role in heart failure prediction for 918 people in the United States. ( For the curious, this set of open data can be found at https://www.kaggle.com/datasets/fedesoriano/heart-failure-prediction .) Columns found in the dataset are as follows. 1. Age: age of the patient [years] 2. Sex: sex of the patient [M: Male, F: Female] 3. ChestPainType: chest pain type [TA: Typical Angina, ATA: Atypical Angina, NAP: Non- Anginal Pain, ASY: Asymptomatic] 4. RestingBP: resting blood pressure [mm Hg] 5. Cholesterol: serum cholesterol [mm/dl] 6. FastingBS: fasting blood sugar [1: if FastingBS > 120 mg/dl, 0: otherwise] 7. RestingECG: resting electrocardiogram results [Normal: Normal, ST: having ST-T wave abnormality (T wave inversions and/or ST elevation or depression of > 0.05 mV), LVH: showing probable or definite left ventricular hypertrophy by Estes' criteria] 8. MaxHR: maximum heart rate achieved [Numeric value between 60 and 202] 9. ExerciseAngina: exercise-induced angina [Y: Yes, N: No] 10. Oldpeak: oldpeak = ST [Numeric value measured in depression] 11. ST_Slope: the slope of the peak exercise ST segment [Up: upsloping, Flat: flat, Down: downsloping] 12. HeartDisease: output class [1: heart disease, 0: Normal] Use R Commander to obtain the most appropriate table to answer the following questions. For each question, copy or take a screenshot of the output in R commander and submit it along with your solutions to each question. To save space, you only need to copy and paste what is asked for in the questions and should adjust the size of the image when appropriate. Hint: construct a frequency distribution in part (a), and contingency tables in parts b through f. (a) If we randomly select a person, what is the probability that they are a female? (2 marks) (b) If we randomly select a person, what is the probability that they are a female and experience exercise-induced angina? (2 marks) (c) If we randomly select a female, what is the probability she has exercise-induced angina? (4 marks) (d) If we randomly select a person with exercise-induced angina, what is the probability that they are a male and are asymptomatic in chest pain? (4 marks) (e) If we randomly select a person, what is the probability they are asymptomatic in chest pain or have no exercise-induced angina, given they are male? (4 marks) 6

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STAT 151 (f) If we randomly select a male with no exercise-induced angina, what is the probability that they have asymptomatic chest pain(4 marks) 3.Linus Ullmark of the Boston Bruins had the best save percentage of any goalie in the 2022- 2023 season. We take this to mean, loosely, that Ullmark has a 93.8% chance of making a save when a puck is shot at his goal. In a practice for the 2023-2024 season, a fellow Bruin will make n=20 shots on Ullmark. HINT: Make sure you use the right p in the problems below. a) What is the probability that Ullmark will miss saving (and hence allow) at most 4 of the shots on goal? (2 marks) p =0.062 (success here is allowing a goal) b) What is the probability that he will save (stop) no more than 17 of the shots on goal? (3 marks) p = 0.938 (success is stopping a goal) c) What is the probability that Ullmark will miss stopping (i.e. allow) at least 2 of the shots on goal and no more than 4 of the shots on goal? (3 marks) Submission Submit your work by accessing the Crowdmark email (or Crowdmark link on mêskanâs) to submit Assignment 2. Please ensure that each picture is properly oriented and easy to read (not fuzzy, not too small, and not taken in a dark room so that it is difficult to read). All work must be submitted to Crowdmark by 6:00 PM on the due date. Avoiding Plagiarism: If you submit an assignment, you are claiming it is your work. Do not allow any part of your work to be copied by anyone else. Where two or more assignments are found to be unreasonably similar, either in whole or in part, and no assistance has been acknowledged, all parties involved are liable to a score of zero on the assignment. MacEwan University’s academic policies are available at: https://www.macewan.ca/contribute/groups/public/documents/policy/academic_integrity.pdf 7

Assignment2_Winter2024 V3

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