Assignment 2_solutions-2

York University Assignment 2 MATH2565 Due Monday, October 30, 2023 at 11:59PM via Crowdmark Instructions: 1. Complete and submit each question in this assignment. 2. Your solution to each question must be submitted to Crowdmark by the due date. No late submissions or non-Crowdmark submissions will be accepted. • You can resubmit as many times as you want prior to the due date. 3. For each question, do your work on a piece of paper and take a photo of your work. Alternatively, you can do your work on an electronic writing surface (such as OneNote). Upload the photo in the appropriate question on Crowdmark. • Crowdmark accepts PDF, JPG, and PNG files. The size limit is 12mb per JPG or PNG file and 25mb per PDF file. 4. For your solutions, you must show your work in order to obtain full marks; little or no credit will be given for an answer without the correct accompanying work. • Calculators are allowed, but you must write out all your calculations in your solutions. • If a question instructs to use R, you only need to provide the final values unless otherwise stated. You must include the graphs generated in R in your submission, if the question asks you to generate it. 5. This assignment is subject to the senate policy on academic honesty and it is the student’s obligation to be familiar of this policy. Submitting this assignment acknowledges that the student understands the polices of academic honesty and rules of this assignment. Question 1 2 3 4 5 6 7 8 Max Score Weight Value 13 12 11 8 8 7 4 5 68 10% 1

1. A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green. 18 of the others are red, and 18 are black. The dealer spins the wheel and, at the same time, rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Round results to 4 decimal places. (a) If you bet on “red”, you win if the ball lands in a red slot. What is the probability of winning? P ( winning ) = P ( red ) = 0 . 4737 (b) Let X be the number of times you won out of the 3 times you bet on “red”. i. What are the possible values of X ? X = 0 , 1 , 2 , 3 where X = 0 means you lost all 3 bets, X = 1 means you won 1 of the 3 bets, X = 2 means you won 2 of the 3 bets, and X = 3 means you won all 3 bets. ii. Find the probability distribution of X . X 0 1 2 3 Probability 0.1458 0.3936 0.3543 0.1063 iii. What are the mean and variance of X ? E ( X ) = 1 . 4211 σ 2 X = 0 . 7479 2

(e) Given that a randomly selected adult gets enough exercise, what is the probability that this adult also gets enough sleep? P ((enough sleep) | (enough exercise) = P ((enough sleep) and (enough exercise)) ((enough exercise)) = 0 . 4 4

3. Statistics Canada reported that in 2020, 17.8% of all students in Canada are in- ternational students. Moreover, 50% of these international students are females whereas 60% of the domestic students are females. (a) Draw a tree diagram. (b) If 1 student is randomly selected, what is the probability that the selected student is a female student? P ( F ) = 0 . 5822 (c) Knowing that a randomly selected student is female, what is the proba- bility that she is a domestic student? P ( D | F ) = 0 . 8471 (d) 2 students are randomly selected. What is the probability that both are female students? P ( FF ) = 0 . 3390 5

5. Let C be the temperature measured in Celsius and F be the temperature measure in Fahrenheit. It is well-known that the relationship between C and F is F = 9 5 C + 32 Round results to 1 decimal place. (a) Toronto’s average daily high temperature in January is 1 . 5 ◦ C with a stan- dard deviation of 5 ◦ C . What is the Toronto’s average daily high temper- ature in January and the corresponding standard deviation expressed in Fahrenheit? It is given that µ C = 1 . 5 and σ C = 5. Therefore, µ F = 34 . 7 and σ 2 F = 81 . 0. Thus, σ F = √ 81 = 9 . 0. (b) Florida’s average daily high temperate in January is 88 ◦ F with a standard deviation of 10 ◦ F . What is Florida’s average daily high temperature and the corresponding standard deviation in January expressed in Celsius? Since F = 9 5 C + 32 = ⇒ C = 5 9 ( F − 32). It is given that µ F = 88 and σ F = 10, so we have µ C = 31 . 1 and σ 2 C = 30 . 8642. Thus, σ C = 5 . 6. 7

6. A club has 30 student members and 10 faculty members. The students are Abel Fisher Huber Moran Reinmann Carson Golomb Jimenez Moskowitz Santos Chen Griswold Jones Neyman Shaw David Hein Kiefer O’Brien Thompson Deming Hernandez Klotz Pearl Utts Elashoff Holland Liu Potter Vlasic and the faculty are Andrews Fernandez Kim Moore Rabinowitz Besicovitch Gupta Lightman Phillips Yang (a) Number each strata such that Abel is 01, Fisher is 02, ... Carson is 06, etc. Obtain a stratified random sample with 6 students and 2 faculty. Use the random numbers table on eClass. For the student stratum, use line 10 and for the faculty stratum, use line 30. List both the number and the student name in your response. Student sample: 17, 13, 01, 29, 07, 14 Names: Kiefer, Jones, Abel, Potter, Golomb, Neyman Faculty sample: 07, 09 Names: Gupta, Phillips (b) Obtain a simple random sample of 3 students using R. Use seed 1026. Include your R script. set.seed (1026) sample(x=1:30 , 3, replace = FALSE) Student sample: 16, 11, 25 Names: David, Chen, Utts 8

8. Multiple sclerosis (MS) is an autoimmune disease that strikes more often the farther people live from the equator. Could vitamin D - which most people get from the sun’s ultraviolet rays - be a factor? Researchers compared vitamin D levels in blood samples from 150 U.S. military personnel who have developed MS with blood samples of nearly 300 who have not. The samples were taken, on average, 5 years before the disease was diagnosed. Those with the highest blood vitamin D levels had a 62% lower risk of MS than those with the lowest levels. (The link was only in whites, not in blacks or Hispanics.) (a) What kind of study was this? Retrospective observational study (b) Who were the subjects? The subjects were U.S. Military personnel, some of whom had developed MS. (c) What were the variables? The variables were the vitamin D blood levels and whether or not the subject developed MS. 10