Assignment 2
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School
York University *
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Course
2565
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
9
Uploaded by SuperMorningNightingale25
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?
(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
?
ii. Find the probability distribution of
X
.
iii. What are the mean and variance of
X
?
2
2. Suppose that 40% of adults get enough sleep, 50% of adults get enough exercise,
and 20% do both.
(a) Are enough sleep and enough exercise disjoint events? Why?
(b) What is the probability that a randomly selected adult gets either enough
sleep or enough exercise but not both?
(c) We want to know the probability that a randomly selected adult gets
enough sleep but not enough exercise.
i. Calculate the probability.
ii. Draw a Venn diagram and shade in the desired probability.
(d) We want to know the probability that a randomly selected adult does not
get enough sleep but gets enough exercise.
i. Calculate the probability.
ii. Draw a Venn diagram and shade in the desired probability.
(e) Given that a randomly selected adult gets enough exercise, what is the
probability that this adult also gets enough sleep?
3
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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?
(c) Knowing that a randomly selected student is female, what is the proba-
bility that she is a domestic student?
(d) 2 students are randomly selected. What is the probability that both are
female students?
4
4. In a recent report in University Affairs, university students in Ontario, on
average, spent
13.50 on lunch with a standard deviation of
7.00. 9 students
are randomly selected from the University of Toronto.
(a) Let
T
be the total lunch spending of these 9 students. What is the mean
and variance of
T
?
(b) Let
¯
X
be the average lunch spending of these 9 students.
What is the
mean and standard deviation of
¯
X
?
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?
(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?
6
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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.
(b) Obtain a simple random sample of 3 students using R. Use seed 1026.
Include your R script.
7
7. In a large city school system with 20 elementary schools, the school board is
considering the adoption of a new policy that would require elementary students
to pass a test in order to be promoted to the next grade. The PTA wants to
find out whether parents agree with this plan. Listed below are some of the
ideas proposed for gathering data. For each, indicate what kind of sampling
strategy is involved.
(a) Put a big ad in the newspaper asking people to log their opinions on the
PTA website.
(b) Randomly select 1 of the elementary schools and contact every parent by
phone.
(c) Send a survey home with every student, and ask parents to fill it out and
return it the next day.
(d) Randomly select 20 parents from each elementary school. Send them a
survey, and follow up with a phone call if they do not return the survey
within a week.
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?
(b) Who were the subjects?
(c) What were the variables?
9
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