2022/5/26 11:26 Quiz: Final 2021SM2 with solution
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Final 2021SM2 with solution
Started: May 26 at 13:26
Quiz Instructions
This is the final exam from 2021SM2.
Any randomly generated question is replaced by one representative question in order to provide
solution to it. In the real exam, students should expect different problems across themselves.
Try to finish by the time provided. Multiple attempts are allowed.
The solution serves as a guideline, not some exam answer template.
Quick Exam Instructions
The format is similar to mid semester tests and practice exams.
NO file upload or pasting from another source.
MUST submit before 6:30pm to avoid late penalty (except for AEA
students)
Detailed instructions are below.
Detailed Exam Instructions
Overview
Open Book Status: Yes.
You are allowed to use notes, textbooks or other resource materials.
This examination contributes 60% to the final subject mark.
This examination paper includes 14 questions and has a total of
72marks.
You are required to answer all questions.
Time
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Writing Time: 3 hrs
Reading Time: 30 minutes
You have 3.5 hours to complete this exam from when you commence
(Except for students who have alternative exam arrangements). This
includes the reading time. Please use a timer to monitor the allocated
time.
Submission
Canvas Quiz will automatically submit the exam when the time limit
(comprised of reading and writing time) has been reached (for
example, after 210 minutes for an exam with 180 minutes of writing
time and 30 minutes of reading time).
Late Penalty
Students must click ‘submit’ at the completion time if they did not start
the exam at the scheduled commencement time to not incur a 10%
late penalty (of the total maximum mark for the exam). For example, a
210-minute exam scheduled at 3:00pm, but commenced at 3:15pm,
will automatically submit at 6:45pm and the student will incur a late
penalty. Students must click 'submit' at 6:30pm to not incur a late
penalty.
Students who are prevented from submitting on time or at all due to
technical difficulties will need to apply for technical consideration with
supporting documentation.
HELP
All exams will be moderated. Please log your concern via the Canvas
chat board in the first instance using the Big Blue Button. Alternatively,
you can call the following numbers for assistance during the exam if
you are experiencing technical difficulties.
Inside Australia: 13MELB (13 6352) (select Option 1 for current
students then select Option 1 again for exam enquiries).
Outside Australia: +61 3 9035 5511 (select Option 1 for current
students and then select Option 1 again for exam enquiries)
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During Exam
Very Important - please do not attempt to upload files or
photos or paste images of any sort into the answer boxes.
Experience has shown that Canvas is unreliable in dealing with
these files and images, they are likely to be lost after submission or
to be corrupted and become unreadable. Uploads / images will
not be marked.
In questions with parts i., ii., iii.,... or (a), (b), (c),...., make sure to
clearly label your answers to match these. The marker needs to see
which part of the question each of your answers relates to. If the
marker can only see a collection of numbers and symbols that don't
clearly relate to the parts of the questions asked, it will not be
possible to give marks.
If you are using notation that has not already been defined in the
question, you must define it yourself.
It is not necessary to create beautiful typeset mathematics with
Greek letters and subscripts and so on. Use simple clear
equivalents that can be typed quickly without the need to search for
different symbols or fonts.
Provide all numerical answers to 3 decimal places unless specified
otherwise.
The time allowed is 3 hours and 30 minutes from when you
commence the Quiz, you can write your answers for the entire time.
Please do not leave it until the very last minute to submit in
case of any pause in the system. Allow a few minutes at the end for
a stress-free submission.
5 ptsQuestion 1
Figure 1-1 shows the histogram of the numbers of Olympic medals for the
participating teams in Tokyo in 2021. Describe any statistical features of the data
based on this histogram.
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Figure 1-2 shows the scatter plot of the numbers of Olympic medals for the
participating teams in London 2012 (x-axis) and Tokyo in 2021 (y-axis). Each dot
represents one participating team. Describe any statistical features of the data
based on this scatter plot.
Figure 1-1: the histogram of the numbers of medals.
Figure 1-2: the scatter plot of the numbers of medals in London and Tokyo
Olympic Games.
Use less than 150 words in total.
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p 0 words
1 ptsQuestion 2
Random variable X may take value 0 or 1; Y may take value 1 or 2.
The following Table shows the joint probability distribution of random variables X
and Y.
Y
1 2
X 0 0.2 0.3
1 0.4
Table: Joint probability distribution of X and Y.
For example, the probability of X=0 and Y=1 is [p01].
What is the value of the blank cell in the table?
(The next question will use the information from this question)
6 ptsQuestion 3
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p 0 words
Use the information from the previous question to calculate the following
quantities.
(a) Marginal probability distribution of X
(b) The expected value, variance and standard deviation of X.
(c) Marginal probability distribution of Y
(d) The expected value, variance and standard deviation of Y.
(e) Conditional probability distribution of Y given X=0.
(f) The conditional expected value, conditional variance and conditional standard
deviation of Y given X=0.

0.5 ptsQuestion 4
There are two assets: bitcoin and pokemon card. After 1 year, the return of the
bitcoin is denoted by B and the pokemon card by C. We know that
(a) B=0.2+0.8Z, where Z is a random variable with a standard normal distribution
N(0, 1).
(b) C=0.1 + 0.7X, where X is a Student-t random variable with degrees of freedom
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6.
(c) Z and X are independent (hence, B and C are independent, too).
(d) A portfolio is constructed as P = wB+(1-w)C, where w=0.4.
What is the portfolio's weight on the pokemon card? ( the next question also needs
condition (a)-(d))
7 ptsQuestion 5
Use the information from the previous question to answer the following questions.
(a) What is the probability that the return of the bitcoin is positive? What are the
expected value and variance of B?
(b) What is the probability that the return of the pokemon card is positive? What
are the expected value and variance of C?
(c) What is the probability that both returns of the bitcoin and pokemon card are
negative after a year?
(d) What are the expected value and variance of the portfolio P? Does this
portfolio P have a lower variance than B and C?
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12pt Paragraph
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p 0 words
3 ptsQuestion 6
p 0 words
Is the following statement True or False? Explain your answer.
For a simple random sample X1, X2, . . . , Xn, its sample mean
¯
X is a consistent estimator for the population mean because its variance
converges to zero if the sample size n increases.

3 ptsQuestion 7
Is the following statement True or False? Explain your answer.
In a hypothesis test, the t-statistic is a random variable.
Edit View Insert Format Tools Table
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Edit View Insert Format Tools Table
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p 0 words
0.5 ptsQuestion 8
A simple random sample is collected to understand the Australians' opinion on
shared bikes. 80 responses are "yes" to the question "would you like to use a
share bike at a small cost (10 cents per km)?" and the remaining 23 answers are
"no". A venture capital (VC) understands that investing in the shared bike project
would be profitable if and only if the population proportion of the "yes" answer to
the survey question is higher than 0.7.
What is the sample size of this survey? (The next question will use the information
in this question.)
10 ptsQuestion 9
Use the information provided from the previous question to apply a statistical test
to investigate if the VC should invest in the shared bike project at 5% significance
level.
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Specifically, you must
(a) define any new notation,
(b) briefly justify your testing procedure,
(c) state the hypotheses,
(d) calculate the test statistic and show your calculation steps,
(e) use the p-value approach,
(f) make decision for the test and briefly explain.
0.5 ptsQuestion 10
The Department of Education and Training Victoria has experimented with a
mobile device control program for children under 12 years old in order to
understand the effect of frequent mobile device usage on children's reading ability.
A simple random sample of 40 children under 12 years old are tested for their
reading ability and the scores are recorded for each child before the program. The
sample mean of the scores is 78, and the sample standard deviation is 24. After
the program, another test is carried out for the same group of children. The sample
mean of the scores is 83, and the sample standard deviation is 21. The sample
standard deviation of the difference of each child's before- and after-program
scores is 15.
What is sample size of this data set? Processing math: 100%
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(The next question will use the information in this question.)
9 ptsQuestion 11
Use the information provided from the previous question to investigate whether the
device control program makes any difference to the mean of children's reading
ability at 1% significance level.
Specifically, you must
(a) define any new notation,
(b) briefly justify your testing procedure,
(c) state the hypotheses,
(d) calculate the test statistic and show your calculation steps,
(e) use the critical value approach,
(f) make decision for the test and briefly explain.
0.5 ptsQuestion 12Processing math: 100%
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The starting salary is one important factor to compare employability across
universities. A simple random sample of 33 students from the University of M
records their average starting annual salaries. The sample mean is $67.4K and
the sample standard deviation is $10.1K. Another simple random sample of 65
students from the University of S report that their average starting annual salary is
$65.2K with a sample standard deviation of $10.2K. ($1K mean 1000 dollars)
What is the total number of observations in this data set?
(The next question will use the information in this question.)
10 ptsQuestion 13
Use the information provided from the previous question to carry out a test to
investigate whether the mean starting salary for students from the University of M
is higher than from the University of S at 10% significance level.
Specifically, you must
(a) define any new notation,
(b) briefly justify your testing procedure,
(c) state the hypotheses,
(d) calculate the test statistic and show your calculation steps,
(e) use the critical value approach,
(f) make decision for the test and briefly explain.
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16 ptsQuestion 14
How is grit related to an individual's overall personal achievement, which includes
income, happiness, health, family and friendship? A government agency has
conducted a survey on a simple random sample of individuals on grit and
summarised an achievement index for them.
A least squares (LS) estimation is applied to estimate the regression model
E Achievementi ∣ Griti = β0 + β1Griti
Some summary statistics are shown in the following table.
Grit Achievement
mean 64.86 34.87
var 456.3 70.10
cov 165.76
sum of squares 409942
sample size (n) 88
sum of squared
residuals
859.65
Use the information to answer all the questions below.
(a) Fill the blanks in the regression table.
table estimate s.e. t p-value
intercept [a] [b] [c] [d]
slope [e] [f] [g] [h]
In your answer, you may arrange the values as in the above output table or simply
write [a]=XXX, ..., [h]=XXX.
(b) What is the value of R2
(c) What is the underlying null hypothesis for β1 from the t-statistic in the Table
of question (a)?
( )
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Not saved
(d) What is the underlying alternative hypothesis for β1from the p-values in the
Table of question (a)?
(e) Compute the 90% confidence interval for β1and explain its interpretation.
Show the steps of calculation.
(f) Carry out a test to investigate if high Grit is associated with high achievement at
1% significance level. You need to
(1) define new notation (if any),
(2) briefly justify your testing procedure,
(3) state the hypotheses,
(4) calculate the test statistic and show your calculation steps,
(5) use the p-value approach,
(6) make decision for the test and briefly explain.
Submit Quiz
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