Assignment 3_Solutions

York University Assignment 3 MATH2565 Due Sunday, November 26, 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 Max Score Weight Value 15 17 4 2 7 13 9 67 10% 1

1. A transport Canada report about air travel found that, nationwide, 76% of all flights are on time. Assume that data regarding flights being on time or not are independent between flights. (a) What is the probability that your next flight is delayed? Given P (on time) = 0 . 76 P (delayed) = 0 . 24 (b) What is the probability that 2 of your next 5 flights are delayed? n = 5 X = # of delayed flights out of 5 flights X ∼ B (5 , 0 . 24) P ( X = 2) = 0 . 2529 (c) What is the probability that at least 2 of your next 5 flights are delayed? X is defined in part (b) P ( X ≥ 2) = 0 . 3461 or P ( X ≥ 2) = 0 . 3461 (d) For the next n flights out of Pearson International airport, how many flights are expected to be on time? What is the standard deviation of flights expected to be on time? Let Y = # of on time flights out of n flights Y ∼ B ( n, 0 . 76) µ Y = E ( Y ) = 0 . 76 n σ Y = √ 0 . 1824 n (e) Let n in part (d) be 100. What is the exact equation for calculating the probability that over 80% of the flights are on time? 2

2. Kateri Vrakking, an Olympic archer, is able to hit the bullseye 80% of the time. Assume each shot is independent of the others. She shoots 10 arrows. (a) Find the mean and variance of the number of bullseyes she may get. Given P (bullseye) = 0 . 80 , n = 10 Let X = # of bullseyes out of 10 X ∼ B (10 , 0 . 8) µ X = E ( X ) = 8 σ 2 X = var ( X ) = 1 . 6 (b) Find the mean and variance of the proportion of bullseye she may get. ˆ p = X 10 = proportion of bullseyes out of 10 µ ˆ p = E (ˆ p ) = 0 . 8 σ 2 ˆ p = var (ˆ p ) = 0 . 016 (c) What is the probability that she does not miss any of her shots? P ( X = 10) = 0 . 1074 (d) What is the probability that there are no more than 8 bullseyes? P ( X ≤ 8) = 0 . 6242 (e) What is the probability that she hits the bullseye more often than she misses? P ( X > 5) = 0 . 9672 (f) Assume she shoots 900 arrows. What is the probability that she hits the bullseye more often than she misses? i. Use R to obtain the exact probability. Provide the R code you used along with the value you obtained. 1 - pbinom (450, 900, 0.8) 4

Therefore, P ( Y > 450) = 1 ii. Obtain the normal approximate of the probability. µ Y = 900(0 . 8) = 720 σ 2 Y = 900(0 . 8)(0 . 2) = 144 P ( Y > 450) = 1 5

4. To assess the accuracy of a kitchen scale, a standard weight known to weigh 1 gram is weighed a total of n times, and the mean, ¯ x , of the weights is computed. Suppose the scale readings are Normally distributed with unknown mean µ and standard deviation σ = 0 . 01 g. How large should n be so that a 99% confidence interval for µ has a margin of error of ± 0 . 0001? n = 66 , 358 7

5. Since previous studies have reported that elite athletes are often deficient in their nutritional intake (for example, total calories, carbohydrates, protein), a group of researchers decided to evaluate Canadian high-performance athletes. A total of 201 athletes from eight Canadian sports centers participated in the study. One reported finding was that the average caloric intake among the 201 women was 2803.7 kilocalories per day (kcal/d). The recommended amount is 2811.5 kcal/d. Assume the population standard deviation of caloric intake among women is 880 kcal/d. Is there evidence that female Canadian athletes are deficient in caloric intake at the 10% significance level? Conduct an appropriate test of significance showing all steps. H 0 : µ ≥ 2811 . 5 H a : µ < 2811 . 5 z = − 0 . 13 p -value = 0 . 4483 p -value > α = 0 . 10 therefore fail to reject H 0 . There is insufficient evidence to conclude that female Canadian athletes are significantly deficient in caloric intake at the 10% significance level. 8

7. A researcher wants to know whether the average time in jail for robbery has increased from what it was several years ago when the average sentence was 7 years. He obtains data on 400 more recent robberies and finds an average time served of 7.5 years. Assume the standard deviation is 3 years. (a) Conduct an appropriate test of significance at the 5% significance level. H 0 : µ ≤ 7 H a : µ > 7 z = 3 . 33 p -value = 0 . 0004 p -value < α = 0 . 05 therefore reject H 0 . There is sufficient evidence to conclude that the average sentence is significantly greater than 7 years at the 5% significance level. (b) What is considered a Type I error in this specific situation? Type I error is to conclude that the average time served is greater than 7 years when in fact µ = 7 years (or µ ≤ 7 years). 10