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Summative Assessment
Statistical Computing and Empirical Methods, Teaching Block 1, 2024
Introduction
This document contains the specification for the summative assessment for the unit Statistical Computing
and Empirical Methods, TB1 2024. Please read carefully the following instructions before you start answering
the questions.
Deadline. Your report is due on 28 November 2024 at 13:00.
Rules: This is an independent task. For the summative assessment you should not share your answers with
your colleagues. The experience of solving the problems in this project will prepare you for real problems in
your career as a data scientist. If someone asks you for the answer, resist!
Support: Whilst this is an independent task, there is a lot of support available if you need it. If you are
unclear about what is required for any part of the assessment then discuss this issue with the our teaching
team in the computer lab or contact your unit director.
Plagiarism: Be very careful to avoid plagiarism. For more details, you should consult the “Academic
Integrity” section under the Assessment tab within the central Blackboard page for the School of Engineering
Mathematics & Technology.
The use of generative AI: The use of generative AI, such as ChatGPT, is prohibited. Any use of generative
AI in this assessment will be considered as plagiarism.
Extenuating circumstances: For more details on the procedure for extenuating circumstances consult the
“Assessment support options” section under the Assessment tab within the central Blackboard page for the
School of Engineering Mathematics & Technology.
Late submission panalty: Coursework that is submitted after a deadline should be subject to a late
submission penalty, unless there is an extension or a justified exceptional circumstance. The more details,
you should consult the central Blackboard page for the School of Engineering Mathematics & Technology or
contact the School office.
Clarity: Clarity is highly important. Be careful to make sure you clearly explain each step in your answer.
You should also include comments within your code when necessary. Your answer should clearly demarcate
which part of the question you are answering. Whenever possible, include pieces of well-written codes in your
report to promote clarity.
Programming language: For Section A of this coursework you should use Tidyverse methods within the
R programming language. For Section B and Section C, you can use either R or Python. Regardless of your
choice of language, it is essential that your answers are clear and well-written.
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Submission points: To submit your solutions, please visit the “Assessment, submission and feedback” tab
on the course webpage at Blackboard. Make sure your submission follows the submission structure described
below.
Multiple submissions: Submitting the coursework multiple times before the deadline is allowed. However,
only the last submission will be considered for marking. You can try to submit an temporary copy before
your final submission if you like.
Submission structure: Please submit a single zip file that contains a folder named “SCEM_???” where “???”
should be replaced by your unique UoB username (e.g., lf22553). The folder should contain three subfolders
named “A”, “B” and “C”.
1 Subfolder "A" should include 1) a PDF file that contains your answers to Section A, and 2) a folder
containing the code and data being used for Section A.
2 Subfolder "B" should include 1) a PDF file that contains your answers to Section B, and 2) a folder
containing the code and data being used for Section B.
3 Subfolder "C" should include 1) a PDF file that contains your answers to Section C, and 2) a folder
containing the code and data being used for Section C.
Time allocation: Section A & B and Section C both contain 50 marks, but we recommend that you allocate
more time for the tasks in Section C, for example 40% on Section A & B and 60% on Section C.
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SECTION A (20 MARKS)
Section A (20 marks)
General instruction: In this part of your assessment, you will perform a data wrangling task using R
programming. Note that clarity is highly important. Be careful to make sure you clearly explain each step in
your answer. You should also include comments within your code when necessary. In addition, make the
structure of your answer clear through the use of headings. You should also make sure your code is clean by
making careful use of Tidyverse methods in R.
(Q1). First download the files entailed "debt_data.csv", "country_data.csv" and "indicator_data.csv" which
are available within the Assessment section within Blackboard.
The file "debt_data.csv" contains debt data for different countries under different indicators, from 1960
to 2023. The indicators are represented by indicator codes (for example, NY.GNP.MKTP.CD). The file
"indicator_data.csv" contains a list of the indicator names as well as their associated indicator codes.
The file "country_data.csv" contains information about the country code, income levels, and regions for
each country.
First, Load the file "debt_data.csv" into an R data frame called "debt_df", load the file "coun try_data.csv" into an R data frame called "country_df", and load the file "indicator_data.csv" into a
data frame called "indicator_df".
Second, use R to check the number of columns and the number of rows that the data frame "debt_df"
has. Display your results.
(Q2). Update "debt_df" by reordering its rows such that the values of the indicator "DT.NFL.BLAT.CD" is
in descending order. Display a subset of the updated "debt_df" consisting of the first 4 rows and the
columns "Country.Code", "Year", "NY.GNP.MKTP.CD", and "DT.NFL.BLAT.CD".
(Q3). In the data frame "debt_df", the indicators are represented by their associated indicator codes rather
than by their names. The data frame "indicator_df" contains a list of indicator names and their
corresponding indicator codes. Create a new data frame called "debt_df2" by combining the data
from the two data frames "debt_df" and "indicator_df". The new data frame "debt_df2" should be
equivalent to "debt_df" except that "debt_df2" now contains indicator names rather than indicator
codes. The indicator names in "debt_df2" should match the indicator codes in "debt_df" according to
their correspondence described in "indicator_df".
Display a subset of "debt_df2" consisting of the first 5 rows and the three columns "Country.Code",
"Year", and "Net financial flows, others (NFL, current US$)".
(Q4). The data frame "country_df" contains information about Region, Income groups, and country name for
each country. Create a new data frame called "debt_df3" by combining data from the two data frames
"debt_df2" and "country_df". The new data frame "debt_df3" should contains a) all columns from
"debt_df2" and b) 3 columns from "country_df" called "Region", "IncomeGroup", and "Country.Name".
Make sure that in each row of "debt_df3", the "Region", "IncomeGroup", and "Country.Name" match
"Country.Code" according to their correspondence described in "country_df".
Your data frames "debt_df3" and "debt_df2" should have the same numbers of rows, but "debt_df3"
has three more columns.
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SECTION A (20 MARKS)
Display a subset of "debt_df3" consisting of the first three rows and 4 columns called "Country.Name",
"IncomeGroup", "Year", and "Total reserves in months of imports".
(Q5). Rename the following 5 columns from their original names to the new names specified below
Original column names New column names
Total reserves in months of imports Total_reserves
External debt stocks, total (DOD, current US$) External_debt
Net financial flows, bilateral (NFL, current US$) Financial_flow
Imports of goods, services and primary income (BoP, current US$) Imports
IFC, private nonguaranteed (NFL, US$) IFC
(Q6). Next generate a summary data frame called “debt_summary” from the data frame “debt_df3” with
the following properties:
Your summary data frame “debt_summary” should contain 7 rows corresponding to the 7 different
Regions, and it should also have 5 columns:
"Region" - the names of the 7 different regions including "East Asia & Pacific", "Europe & Central Asia"
etc.
"TR_mn" - the average of "Total_reserves" in each region.
"ED_md" - the median of "External_debt" in each region.
"FF_quantile" - the 0.2 quantile of "Financial_flow" in each region.
"IFC_sd" - the standard deviation of "IFC" in each region.
All missing values should be discarded when computing the summary data.
(Q7). Based on your data frame “debt_df3”, create a violin plot of "Financial_flow" for each of the regions.
The violin plots should be displayed in the same figure and with different colors representing different
regions. Ignore all missing values and all values that are smaller than −108 or bigger than 108
.
Your plot is expected to look as follows.
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SECTION A (20 MARKS)
East Asia & Pacific
Europe & Central Asia
Latin America & Caribbean
Middle East & North Africa
South Asia
Sub−Saharan Africa
−1e+08 −5e+07 0e+00 5e+07 1e+08
Financial Flow
Region
East Asia & Pacific
Europe & Central Asia
Latin America & Caribbean
Middle East & North Africa
South Asia
Sub−Saharan Africa
(Q8). Based on the data frame “debt_df3”, create a plot which displays the "Total_reserves" as a function
of the years (from 1960 to 2023), for each of the following countries: Italy, France, United Kingdom,
Sudan, Afghanistan, and Brazil. Additionally, the values of "Total_reserves" should be displayed in
different panels according to the income groups of the countries. Use different colors to represent
different countries.
Your plot is expected to look as follows.
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Region
SECTION A (20 MARKS)
High income Low income Upper middle income
197019801990200020102020197019801990200020102020197019801990200020102020
0
5
10
15
Year
Country Name
Afghanistan
Brazil
France
Italy
Sudan
United Kingdom
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Total Reserves
SECTION B (30 MARKS)
Section B (30 marks)
B.1
Suppose a product is being sold in a supermarket. We are interested in knowing how quickly the product
returns to the shelf again after it is sold out. Let X be a continuous random variable denoting the length of
time between the time point at which it is sold out and the time point at which it is placed on the shelf again.
So X should be a non-negative number, and X = 0 means that the product gets on the shelf immediately
after it is sold out. Here, we assume that the probability density function of X is given by
pλ(x) = ( ae−λ(x−b)
if x ≥ b,
0 if x < b,
where b > 0 is a known constant, λ > 0 is a parameter of the distribution, and a is to be determined by λ
and b.
(1) First, determine the value of a: derive a mathematical expression of a in terms of λ and/or b.
(2) Derive a formula for the population mean and standard deviation of the random variable X with
parameter λ.
(3) Derive a formula for the cumulative distribution function and the quantile function for the random
variable X with parameter λ.
(4) Suppose that X1, · · · , Xn are independent copies of X with the unknown parameter λ > 0. What is
the maximum likelihood estimate λMLE for λ?
Now download the .csv file entitled “supermarket_data_2024” from the Assessment section within Blackboard.
The .csv file contains data on the length of time (in seconds) taken by a product to get on the shelf again
after being sold out. So the sample is a sequence of time lengths. Let’s model the sequence of time lengths in
our sample as independent copies of X (X is the random variable mentioned above) with parameter λ and
known constant b = 300 (seconds). Answer the following questions (5) and (6).
(5) Given the sample, compute and display the maximum likelihood estimate λMLE of the parameter λ.
(6) Apply the method of Bootstrap confidence interval to obtain a confidence interval for λ with a confidence
level of 95%. To compute the Bootstrap confidence interval, the number of resamples (i.e., subsamples
that are generated to compute the bootstrap statistics) should be set to 10000.
Next, conduct a simulation study to explore the behaviour of the maximum likelihood estimator:
(7) Conduct a simulation study to explore the behaviour of the maximum likelihood estimator λMLE for λ
on simulated data X1, · · · , Xn (as independent copies of X with parameter λ) according to the following
instructions. Let b = 0.01 and the true parameter be λ = 2. Generate a plot of the mean squared error
as a function of the sample size n. You should consider sample sizes from 100 to 5000 in increments of
10. For each sample size, consider 100 trials. In each trial, generate a random sample X1, · · · , Xn (as
independent copies of X with parameter λ = 2), and then compute the maximum likelihood estimate
λMLE for λ based upon the sample. Display a plot of the mean square error of λMLE as an estimator
for λ as a function of the sample size n.
B.2
Consider a bag of a red balls and b blue balls (the bag has a + b balls in total), where a ≥ 1 and b ≥ 1. We
randomly draw two balls from the bag without replacement. That means, we draw the first ball from the bag
and, WITHOUT returning the first ball to the bag, we draw the second one. Each ball has an equal chance
of being drawn. Now we record the colour of the two balls drawn from the bag, and let X denote the number
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B.2 SECTION B (30 MARKS)
of red balls minus the number of blue balls. So X is a discrete random variable. For example, if we draw one
red ball and one blue ball, then X = 0. Answer the following questions from (1) to (11).
(1) Give a formula for the probability mass function pX : R → [0, 1] of X.
(2) Use the probability mass function pX to obtain an expression of the expectation E(X) of X (i.e., the
population mean) in terms of a and/or b.
(3) Give an expression of the variance Var(X) of X in terms of a and b.
(4) Write a function called compute_expectation_X that takes a and b as inputs and outputs the expectation
E(X). Write a function called compute_variance_X that takes a and b as input and outputs the variance
Var(X). Display your code.
In the following questions, we additionally assume that X1, X2, · · · , Xn are independent copies of X. So
X1, X2, · · · , Xn are i.i.d. random variables having the same distribution as that of X. Let X = n
1 P n
i=1 Xi
be the sample mean.
(5) Give an expression of the expectation of the random variable X in terms of a, b.
(6) Give an expression of the variance of the random variable X in terms of a, b and n.
(7) Create a function called sample_Xs which takes as inputs a, b and n and outputs a sample
X1, X2, · · · , Xn of independent copies of X.
(8) Let a = 3, b = 5 and n = 100000. First, compute the numerical value of E(X) using the
function compute_expectation_X and compute the numerical value of Var(X) using the function
compute_variance_X. Second, use the function sample_Xs to generate a sample X1, X2, · · · , Xn of
independent copies of X. With the generated sample, compute the sample mean X and sample
variance. How close is the sample mean X to E(X)? How close is the sample variance to Var(X)?
Explain your observation.
Moreover, let µ := E(X) and σ := p Var(X)/n (the random variable X is defined above), and let fµ,σ : R →
[0, ∞) be the probability density function of a Gaussian random variable with distribution N (µ, σ2
), i.e., the
expectation is µ and the variance is σ
2
. Next, conduct a simulation study to explore the behaviour of the
sample mean X by answering questions (9)-(11).
(9) Let a = 3, b = 5 and n = 100. Conduct a simulation study with 50000 trials. In each trial, generate a
sample X1, · · · , Xn of independent copies of X. For each of the 50000 trials, compute the corresponding
sample mean X based on X1, · · · , Xn.
(10) Create a scatter plot of the points {(xi
, fµ,σ(xi))} where {xi} are a sequence of numbers between µ−3σ
and µ + 3σ in increments of 0.1σ. Then append to the scatter plot a curve representing the kernel
density of the sample mean X within your simulation study (with 50000 trials). Use different colours
for the point {(xi
, fµ,σ(xi))} and the density curve of the sample mean X.
(11) Describe the relationship between the density of X and the function fµ,σ displayed in your plot. Try to
explain the reason.
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SECTION C (50 MARKS)
Section C (50 marks)
In this part of the assessment, you are asked to complete a Data Science report which demonstrates your
understanding of a statistical method. The goal here is to choose a topic that you find interesting and explore
that topic in depth. You are free to choose a topic and data set that interests you.
There will be an opportunity to discuss and get advice on your chosen direction in the computer labs.
Below are two flexible example structures you can consider for this section of your report. If you are unsure
what to do, choose one of the following. Note that you should not submit more than one of the example tasks
below.
Example task 1
Investigate a particular hypothesis test e.g. a Binomial test, a paired Student’s t test, an unpaired Student’s
t test, an F test for ANOVA, a Mann-Whitney U test, a Wilcoxon signed-rank test, a Kruskal Wallis test, or
some other test you find interesting.
Note that clarity of presentation is highly important. In addition, you should aim to demonstrate a depth of
understanding. For this hypothesis test you are asked to do the following:
1. Give a clear description of the hypothesis test being considered, including the details of the test statistic
and p-value, the underlying assumptions, the null hypothesis and the alternative hypothesis. Give
an intuitive explanation for why the test statistic is useful in distinguishing between the null and the
alternative.
2. Perform a simulation study to investigate the probability of type I error under the null hypothesis for
your hypothesis test. Your simulation study should involve randomly generated data which conforms
to the null hypothesis. Compare the proportion of rounds where a Type I error is made with the
significance level of the test. What happens when a different significance level is used?
3. Choose a suitable real-world data set (for example, some places to find data sets are described below).
Ensure that your chosen data set is appropriate for your chosen hypothesis test. For example, if your
chosen hypothesis test is an unpaired t-test then your chosen data set must have at least one continuous
variable and contain at least two groups. It is recommended that your data set for this task not be too
large. You should explain the source and the structure of your data set within your report. You should
also explain the related problem on which you want to perform the test.
4. Carefully discuss the appropriateness of your statistical test in this setting and how your hypotheses
correspond to different aspects of the data set. You may want to use plots to demonstrate the validity
of your underlying assumptions. Draw a statistical conclusion and report the value of your test statistic,
the p-value and a suitable measure of effect size.
5. Discuss what scientific conclusions you can draw from your hypothesis test. Discuss how these would have
differed if the result of your statistical test had differed. Discuss key experimental design considerations
necessary for drawing any such scientific conclusion. For example, perhaps an alternative experimental
design would have allowed one to draw a conclusion about cause and effect?
6. Exploring further this hypothesis test on one topic/direction of your choice. This could be for example
discussing a property of the test such as how the power of the chosen test changes with sample size,
significance level, or effect size. As another example, how robust is the test when assumptions are
violated and is there a robust alternative? How does the test compare to its non-parametric alternatives?
How does the frequentist test compare with its Bayesian alternative? These are just a few examples.
Make a clear statement on the question of interest and your conclusions. The details of your approach
to support your findings should be visible within your report, and experiments or simulation studies
can be included if needed.
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Example task 2 SECTION C (50 MARKS)
Example task 2
Investigate a particular method for supervised learning. This could either be a method for regression or
classification but should be a method with at least one tunable hyperparameter. You could choose one from
ridge regression, k-nearest neighbour regression, a regression tree, regularized logistic regression, k-nearest
neighbour classification, a decision tree, a random forest or another supervised learning technique you find
interesting.
Note that clarity of presentation is highly important. In addition, you should aim to demonstrate a depth of
understanding.
1. Give a clear description of the supervised learning technique you will use, including the underlying
principles and any assumptions. Explain how the training algorithm works and how new predictions
are made on test data. Discuss what type of problems this method is appropriate for.
2. Choose a suitable data set where this method can be applied. Perform a train, validation, and test split
(for example, some places to find data sets are described below). Be careful to ensure that your data set
is appropriate for your chosen algorithm. For example, if you have chosen to investigate a classification
algorithm then your chosen data set must contain at least one categorical variable. Your data set for
this task does not need to be large to obtain good results. The size of your data set should not exceed
100MB and you should aim to use a data set well within this limit. Your report should carefully give
the source for your data. In addition, describe your data set. How many features are there? How many
examples? What type is each of the variables (e.g. categorical, ordinal, continuous, binary etc.)? You
should also explain the associated problem that you will solve using your supervised learning method.
3. What is an appropriate metric for the performance of your model? Give a clear explanation of the
metric. Explore how the performance of your model varies on both the training data and the validation
data as you vary the amount of training data used. You should compare the performance of the models
across different sizes of the training data.
4. Explore how the performance of your model varies on both the training data and the validation data as
you vary a hyperparameter.
5. Choose a hyper-parameter and report your performance based on the test data. Can you get a better
understanding by using cross-validation?
6. Exploring further this supervised learning method on one topic/direction of your choice. This could
be for example discussing how the bias-variance trade-off impacts the performance of the chosen
method. As another example, is your model robust? How does the performance of the method change
when applied to imbalanced datasets? Does your method work on small data and if not is there
an suitable alternative? You could also investigate how different regularisation techniques affect the
model’s performance, or carefully compare the chosen method with other methods. These are just a
few examples. Make a clear statement on the question of interest and your conclusions. The details
of your approach to support your findings should be visible within your report, and experiments or
simulation studies can be included if needed.
Further instruction for Section C.
Note:
1. Do not complete and submit more than one of the above tasks. These are example tasks and you should
only choose one. The goal here is to explore a topic in detail.
2. You will be graded on the level of understanding of the key concepts demonstrated within your report.
Additional marks will be given for more advanced methods, provided that a very strong level of
understanding is displayed. However, you should avoid choosing complex methods without properly
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Further instruction for Section C. SECTION C (50 MARKS)
demonstrating your understanding. The main focus here is a clear understanding and you should
not sacrifice understanding for the sake of complexity. A clear understanding of the basic concepts is
paramount.
3. You do not need to use large data sets. The dataset you choose should not be larger than 100MB. This
is an upper bound. You should aim to use a data set well within this limit.
4. We expect that your approach should be visual and clear within the report itself. Therefore it is highly
recommended to include pieces of clear and well-written code along with necessary comments and
explanations within the report itself.
5. We expect that you interpret and make sense of the experiment results obtained, instead of displaying
a list of the results without explanation or analysis. A high quality report should be able to use the
experimental results to support its conclusions and findings in a consistent manner.
6. We do not have a page limit for the report. A rough guideline is that your report should ideally be no
more than 10 pages, if all figures and large pieces of code were removed. However, this is not a strict
constraint. Again, clarity is highly important, and you should include sufficient details to demonstrate
your approach and the level of understanding of the key concepts.
Data sets
There are a vast number of freely available data sets across the internet. Below are a few example sources.
You are also welcome to use data sets from other sources. Any data you use should be freely available and
accessible. The source of your data and the steps required to retrieve it should also be described within your
main report.
You should also explain its structure e.g. the number of rows and the number of columns, and what the data
in each column of interest represent for, · · ·. You are encouraged to use tabular data throughout.
Final remarks
Throughout your report you should emphasise:
• Reproducible analysis (be careful with randomised procedures).
• Clear and informative visualisations of your results.
• Demonstrate a depth of understanding.
• A clear writing style.
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