STA238 - Probability, Statistics and Data Analysis II
Winter Midterm
March 16, 2022
10:10 AM - 11:40 AM ET, Submit by 12:00 PM ET
This question document contains 7 pages (including this cover page) and 5 problems.
The midterm is an individual and independent open-book test. The expectation is the problems
below are completed by hand with your work shown (e.g. integration steps, etc.). Your work must be
submitted on Crowdmark no later than 12 PM ET. This is a strict deadline. The term ‘open-book’
means you have access to the following resources within our course ONLY:
• Course notes
• Course Textbooks
• Probability calculations in R (not mandatory), scientific calculator, probability tables (found
on our Midterm Information Page)
Using any other resources beyond our course is considered an unauthorized aid. This includes but
is not limited to:
• Posting on discussion board or elsewhere for help (e.g., Slack, Discord, Facebook, etc.)
• Working with a classmate (this is an individual and independent assessment!). The work you
submit must be entirely your own
• Receiving assistance, tips, hints, or otherwise from someone else
• Submitting any work that is not your own
• Searching for answers on other platforms (e.g. external textbooks, TAs, Google, etc.)
u Clearly state and define any variables
and/or distributions used, including pa-
rameters . Use appropriate probability and
event notation in your work.
u Show all your work, including calcula-
tions, integrations, etc. We can only assess
when you can show us
u Round all final answers to 4 decimal places (or
2 decimal places if expressed as a percentage).
u Send any technical issues or urgent ques-
tions that occur during the test to the
course email (sta238@utoronto.ca)!
Problem Points Score
1 11
2 3
3 8
4 8
5 9
Total: 39

1. (11 points) You examine historical data and find that the wait time for a common service
has a Gamma distribution with ↵ = 1.5 and = 2. The graph of the density function for a
Gamma(↵ = 2, = 2) is plotted below, with the red vertical line indicating the mean of the
density function.
Note: If X ⇥ Gamma(↵, ), E(X) = ↵, V (X) = ↵2
(a) (3 points) TRUE/FALSE and explain your choice: According to Central Limit Theorem,
X30 will have an approximately normal distribution.
(b) (5 points) What is the probability that you will observe a sample mean of 30 observations
that is within 0.5 of the true mean? Justify your calculation and explicitly state the
distribution used.
STA238 - Probability, Statistics and Data Analysis II Page 2 of 7
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(c) (3 points) What information about the interval in (b) does Chebyshev’s inequality give
you? Calculate it.
2. (3 points) The function is designed to return what output? Write your selection here:
Briefly explain your choice.
a) Density histogram of sample variance for a dataset of size n
b) Density histogram of simulated data from an unknown distribution
c) Density histogram of simulated data from a known distribution
d) Simulated data points from some distribution
STA238 - Probability, Statistics and Data Analysis II Page 3 of 7
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3. (8 points) It is important that face masks used by firefighters be able to withstand high tem-
peratures because they commonly work in temperatures of 90 to 250 degrees Celsius. In a test
of one type of mask, 11 of the 55 masks had the lenses pop out at 120 degrees Celsius.
(a) (5 points) Construct a 92% confidence interval for the true proportion of masks of this
type that will fail at 120 degrees Celsius in this way. Use the method that results in the
most conservative interval.
(b) (3 points) Do you think the interval you’ve constructed has exactly 92% confidence? If
not, do you think it is higher or lower? Explain. Note: Your response should not rely on
any calculations, but instead provide a rational justification for your response.
STA238 - Probability, Statistics and Data Analysis II Page 4 of 7
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4. (8 points) Below is a data set on testosterone measurements on 22 male athletes, sorted in
numerical order. Use this data to construct a fully labeled modified boxplot representing the
distribution of testosterone among these 22 male athletes. Use the percentile calculation method
for any percentiles you need to compute. How would you describe the shape of the distribution?
Note: The [#] at the beginning of each line indicates the element number starting
in that row. For example, 190.0 is the 11th sorted data point.
STA238 - Probability, Statistics and Data Analysis II Page 5 of 7
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5. (9 points) Let X1, X2, ..., X8 be a random sample from a distribution with parameter ✓. The
density function is given by:
f(x∂✓) = ~ÑÑÑÑÇÑÑÑÑÄ
✓3
2
x3e✓x, x > 0
0, otherwise
Note E(X) = 3
✓
and V (X) = 3
✓2
(a) (2 points) Write out the likelihood and log-likelihood function of ✓.
(b) (5 points) Derive the maximum likelihood estimator for ✓. Find also the maximum likeli-
hood estimator for the variance of X. What property allows you to do this easily?
STA238 - Probability, Statistics and Data Analysis II Page 6 of 7
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(c) (2 points) Your random sample yielded the following observations:
0.78 0.60 0.89 0.63
0.28 0.65 0.36 1.16
Find the maximum likelihood point estimate for ✓ and for E(X).
STA238 - Probability, Statistics and Data Analysis II Page 7 of 7
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