1131-ass2 (1)
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York University *
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Course
1131
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
3
Uploaded by CountFlowerSpider39
MATH1131 Assignment 2
Q1(6).
Parents were asked if sports are equally important for boys and girls.
Suppose that the proportion
of parents who agree the equal importance in the population is actually 0.65. In a survey, of the 400 parents
interviewed, 70% agreed that boys and girls should have equal opportunities to participate in sports.
1(2).
Describe the sampling distribution of the sample proportion of parents who agree that boys and girls
should have equal opportunities.
2(4). What is the probability of observing a sample proportion as large as or larger than the observed value ˆ
p
= 0.70?
Q2(10). Adults is recommended to get a booster dose of the Td (tetanus and diphtheria) vaccine every 10 years.
Suppose the length of time (in years) that the vaccine provides protection has an exponential distribution with
a mean of 15 years.
1(2). Find the value of
λ
for this distribution.
2(4). Find the probability that a randomly selected person still has vaccine-induced immunity 10 years after
receiving their shot.
3(4). Suppose two randomly selected people receive their booster on the same day. What is the probability
that both individuals still have vaccine-induced immunity five years after receiving their booster?
Q3(10).
A local gym states that their clients spend on average of 60 minutes in the gym, with a standard
deviation of 20 minutes. Assume the time spent in the club follows a normal distribution.
1(4). Find the probability that one randomly selected client spends more than 80 minutes in the gym.
2(4). Consider a random sample of 16 clients, find the probability that the sample average time spent in the
club is between 56 mins and 68 mins, inclusive.
3(2). Do you think the normal distribution is an appropriate model for the time spent in the gym? This is an
open question, just state your opinion.
Q4(6).
Use R to complete the following.
R code and R output is requested for solution.
R markdown is
recommended.
Please keep you solution clear and easy to read.
Unclear solution will not be graded.
0 for
manual solution
1(3). Generate a random sample with sample size
n
= 100, from standard normal distribution. Calculate the
mean, the standard deviation, and create a histogram for the random sample generated.
2(2). Calculate the probability:
X
∼
N
(6
,
7
2
),
P
(3
≤
X
≤
13)
.
3(1). Calculate the probability:
X
∼
B
(164
,
0
.
234),
P
(
X
= 35)
.
Q5(16). Using the applet link
https://www.rossmanchance.com/applets/2021/confsim/ConfSim.html
and set left panel parameters: ”Statistic = mean” and ”Method = z with sigma”. You can change any other
parameters to obtain 6 different results in the plots.
1(6). Take screenshots and submit each of your six results. Each of your screenshot should include all of the
items in my screenshot below.
1
2(6). There are three plots shown in each screenshot. I will label them A, B, and C. Plot B is called ”Most
Recent Sample” and plot C is called ”Sample statistics (CI midpoints)”. Write 2-4 sentences explaining what
appears in each plot.
3(4). Referring to your six screenshots from part 1, write 100-200 words explaining how confidence intervals
work.
Q6(8). Many scientists are concerned about changes in the ice thickness due to global warming. Suppose the
historical mean thickness of the ice just off the coast of Barrow, Alaska, is 3.32 meters. A recent random sample
(
n
= 28) of the ice thickness near Barrow was obtained. The sample mean ice thickness is 4.036 meters. Assume
σ
= 2
.
8 meters, and the underlying distribution is normal. Conduct a hypothesis test to determine whether
there is any change in the mean ice thickness.
Q7(12). Manatees are one creature found in coastal waterways. 10 years ago, researchers concluded that the
true mean weight of manatees was approximately 1050 lbs. This year, a random sample of 12 manatees were
selected, the sample weights are: 956, 1012, 954, 988, 973, 1048, 1075, 1064, 856, 1026, 1031, 1064. Assume the
underlying distribution of manatee weights is normal, and population standard deviation
σ
is unknown.
1(8). Implement a hypothesis test to verify if there is any evidence to suggest the true mean weight decreases
now.
2(4). Use the given sample data to calculate 90% confidence interval for current mean weight of manatees.
Q8(8). Suppose that you are given a random sample (
n
= 18) of annual salaries for a specific position. The
observed sample mean is
x
=
54,500. Assume population is normal with standard deviation
σ
= 9
,
500.
1(4) Find a 90% confidence interval for the true mean annual salary.
2(4) How large the sample size is necessary for the margin of error to be at most
3000?
Q9(12). Historical data shows that approximately 64% of Ontario public school principals in one province were
female. Recently, a study showed that 231 of 328 randomly selected principals in Ontario were female.
1(6). Is there any evidence to suggest that the proportion of female principals has changed? Use significance
testing with
α
= 0
.
05
.
2(3). Find the 99% confidence interval for the current true population proportion
p.
3(3). Using the above information as prior experience, find the minimum sample size such that margin of error
is no larger than 0.1 for a 99% confidence interval.
Q10(9).Use R to complete the following.
R code and R output is requested for solution.
R markdown is
recommended.
Please keep you solution clear and easy to read.
Unclear solution will not be graded.
0 for
manual solution.
You are given the data set
4.52, 1.02, 0.27, 10.38, 13.04, -4.10, 8.21, -0.64, 4.35, 2.74,
2
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