STA261H1S Term Test 2 (Wed Evening)
March 30th, 2022 7:10pm - 9:10pm EST
All relevant work must be shown for full marks
Total Marks = 50
Need to show all your work to receive full marks
Note: There are 5 questions (with parts) and you have to answer all of them to receive full marks.
The answers can be hand written or typed. You can submit multiple files but please make sure all
the answers are submitted in pdf, docx, jpg or png files. The questions are created in a way such
that all of them can be answered within 1 hour and 45 minutes, so that you have 15 minutes to
scan and submit. Please make sure that you leave yourself enough time to scan and submit your
work.
ACADEMIC INTEGRITY: This is an open book exam. However, the University of Toronto
treats cases of plagiarism and cheating very seriously. It is the students’ responsibility to know
the content of the University of Toronto’s Code of Behaviour on Academic Matters. All sus-
pected cases of academic dishonesty will be investigated following procedures outlined in the above
document. If you have questions or concerns about what constitutes appropriate academic be-
haviour or appropriate research and citation methods, you are expected to seek out additional
information on academic integrity from your instructor or from other institutional resources (see
http://academicintegrity.utoronto.ca/). Here are a few guidelines regarding academic in-
tegrity:
• You may consult class notes/lecture slides during the test, however sharing or discussing
questions or answers with other students is an academic offense.
• Students must complete all assessments individually and independently. Working together is
not allowed.
• Paying anyone else to complete your assessments for you is serious academic misconduct.
• Sharing your answers/work/code with others is academic misconduct.
• Looking up solutions to test/quiz problems online and copying what you find is an academic
offense.
• All work that you submit must be your own! You must not copy mathematical derivations,
computer output and input, or written answers from anyone or anywhere else. Unacknowl-
edged copying or unauthorized collaboration will lead to severe disciplinary action, beginning
with an automatic grade of zero for all involved and escalating from there. Please read the
UofT Policy on Cheating and Plagiarism, and don’t plagiarize.
Please don’t upload this document on any social media platforms, chegg, slideshare or
coursehero. Uploading this document to any such website will be treated as a serious
academic offense and we will take actions based on University of Toronto’s policies
regarding plagiarism. We will monitor these websites closely to detect such incidents.
Best of luck!
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Question 1
(a) [5 Marks] A two sided 95% confidence interval for µ in the location Normal model N(µ, σ20),
calculated from a an observed sample is (54, 66). ( Note: sample size is not given to you)
Calculate the lower limit of the 90% confidence interval for µ, using the data from the same
sample.
(b) [8 Marks] Using R, I generated a sample of 49 observations from a Poisson(λ) distribution and
calculated an approximate two sided 95% confidence interval for λ using the approximation
based on the central limit theorem. The lower limit of this confidence interval is L = 1.64.
Calculate the upper limit of this confidence interval.
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Question 2
Suppose a sample of n = 10 employees were collected from a company and their income (in
$1000) was reported. The incomes were s = 4.00, 5.26, 4.84, 6.77, 5.23, 5.64, 3.84, 6.43, 3.35, 4.28 ∼
N(µ, σ2), with both µ and σ unknown.
(a) [5 Marks] Estimate the population average income µ and the 80% confidence interval.
(b) [4 Marks] Perform a two sided test to test H0 : µ = 6 at α = 0.1.
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Question 3
Suppose that x = (1, 0, 1, 0, 1) is an observed sample from Bernoulli(θ) distribution and the prior
distribution of θ has p.d.f.:
pi(θ) =
{
4θ3, if 0 < θ < 1
0 otherwise.
(a) [5 Marks] Calculate the posterior expectation of θ.
(b) [5 Marks] Calculate the posterior variance of 1θ .
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Question 4
(a) [5 Marks] I generated a sample from a location Normal model with σ20 = 400. The two sided
95% confidence interval for the mean (µ) calculated using this sample I generated is (60, 70).
This interval has all the information necessary to test the null hypothesis H0 : µ = 62 against
the alternative H1 : µ > 62. Calculate the value of the observed z-statistic and the p-value
for the z-test of the null hypothesis H0 : µ = 62 against the alternative H1 : µ > 62. Assume
α = 0.05
(b) [3 Marks] Calculate the number of observations in this study.
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Question 5
A Biologists is interested in the proportion, θ of badgers in a particular area which carry the infection
responsible for bovine tuberculosis. The prior distribution of θ is Beta(1, 19). The biologists
captures a random sample of 20 badgers and tests them for infections. The observed number of
infections were 2. Assume that the
(a) [4 Marks] Find the prior mean and prior mode of θ.
(b) [6 Marks] Find the posterior mean and posterior mode for the distribution of θ.
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