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Data Structures & Algorithms

Find the missing integer from a array of consequtive integers

Find the Missing Number

Write a function moving_window to find the moving window average

Moving Window

Given a dictionary consisting of many roots and a sentence, write a function to stem all the words in the sentence with the root forming it.

Replace Words with Stems

Given an array and a target integer, write a function that returns the indices of two integers in the array that add up to the target integer.

Target Indices

Write a function to rotate an array by 90 degrees in the clockwise direction.

Matrix Rotation

Machine Learning

Interpret coefficients of logistic regression for categorical and boolean variables

Coefficients of Logistic Regression

How do we deal with the missing square footage data to construct our model

Missing Housing Data

How would we build a model to detect fraud and text customers to approve or deny fraudulent transactions

Bank Fraud Model

Lasso vs Ridge

Explain the key differences between classification models and regression models

Classification and Regression

Behavioral

Describing a data project and its challenges

Hurdles In Data Projects

Strategically resolving misaligned expectations with stakeholders for a successful project outcome

Stakeholder Communication

Prioritized debt reduction, process improvement, and a focus on maintainability for fintech efficiency

Decreasing Tech Debt

How to present complex data insights with clarity and adaptability tailored to a specific audience

Presentations and Insights

How would you answer when an Interviewer asks why you applied to their company?

Why Do You Want to Work With Us

Calculate the first touch attribution for each `user_id` that converted.

First Touch Attribution

Write a query to return whether each user's subscription date range overlaps with any other completed subscription

Subscription Overlap

Get the top 3 highest employee salaries by department

Top Three Salaries

User Experience Percentage

Find how many users logged in a certain number of times on a given day

Daily Logins

Analytics

We want to figure out if users are creating multiple accounts to upvote their own comments.

Fraudulent Upvotes

Create and write queries for health metrics for stack overflow 

Community Health Metrics

Achieving maximum profits through dynamic pricing with real-time supply and demand fluctuations

Understanding Dynamic Pricing Strategy

How would you increase the user engagement of a certain demographic?

Romantic Reduction

What kind of analysis would you conduct to recommend changes to the UI?

User Journey Analysis

How would you build a database for a consumer file storage company like Dropbox?

Note: The database must be able to store many different file type formats and keep updates and history of file changes.

How would you build a database for a consumer file storage company like Dropbox

Dropbox Database

Data Modeling

Given the <code>employees</code> and <code>departments</code> table, write a query to get the top 3 highest employee salaries by department. If the department contains less that 3 employees, the top 2 or the top 1 highest salaries should be listed (assume that each department has at least 1 employee). 

Note: The output should include the full name of the employee in one column, the department name, and the salary. The output should be sorted by department name in ascending order and salary in descending order. 

Example:

Input:

<code>employees</code> table

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>first_name</code></td>
<td>VARCHAR</td>
</tr>

<tr>
<td><code>last_name</code></td>
<td>VARCHAR</td>
</tr>

<tr>
<td><code>salary</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>department_id</code></td>
<td>INTEGER</td>
</tr>
</tbody>
</table>
<code>departments</code> table

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>name</code></td>
<td>VARCHAR</td>
</tr>
</tbody>
</table>
Output:

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>employee_nam</code>e</td>
<td>VARCHAR</td>
</tr>

<tr>
<td><code>department_name</code></td>
<td>VARCHAR</td>
</tr>

<tr>
<td><code>salary</code></td>
<td>INTEGER</td>
</tr>
</tbody>
</table>

Given a table of product subscriptions with a subscription start date and end date for each user, write a query that returns true or false whether or not each user has a subscription date range that overlaps with any other completed subscription.

Completed subscriptions have <code>end_date</code> recorded.

Example:

Input:

<code>subscriptions</code> table

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>user_id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>start_date</code></td>
<td>DATETIME</td>
</tr>

<tr>
<td><code>end_date</code></td>
<td>DATETIME</td>
</tr>
</tbody>
</table>

<table>
<thead>
<tr>
<th><code>user_id</code></th>
<th><code>start_date</code></th>
<th><code>end_date</code></th>
</tr>
</thead>

<tbody>
<tr>
<td>1</td>
<td>2019-01-01</td>
<td>2019-01-31</td>
</tr>

<tr>
<td>2</td>
<td>2019-01-15</td>
<td>2019-01-17</td>
</tr>

<tr>
<td>3</td>
<td>2019-01-29</td>
<td>2019-02-04</td>
</tr>

<tr>
<td>4</td>
<td>2019-02-05</td>
<td>2019-02-10</td>
</tr>
</tbody>
</table>
Output:

<table>
<thead>
<tr>
<th><code>user_id</code></th>
<th><code>overlap</code></th>
</tr>
</thead>

<tbody>
<tr>
<td>1</td>
<td>1</td>
</tr>

<tr>
<td>2</td>
<td>1</td>
</tr>

<tr>
<td>3</td>
<td>1</td>
</tr>

<tr>
<td>4</td>
<td>0</td>
</tr>
</tbody>
</table>

The schema below is for a retail online shopping company consisting of two tables, <code>attribution</code> and <code>user_sessions</code>.

<ul>
<li>The attribution table logs a session visit for each row.</li>

<li>If <code>conversion</code> is <code>true</code>, then the user converted to buying on that session.</li>

<li>The <code>channel</code> column represents which advertising platform the user was attributed to for that specific session.</li>

<li>Lastly the <code>user_sessions</code> table maps many to one session visits back to one user.</li>
</ul>

First touch attribution is defined as the channel with which the converted user was associated when they first discovered the website.

Calculate the first touch attribution for each <code>user_id</code> that converted. 

Example:

Input:

<code>attribution</code> table

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>session_id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>channel</code></td>
<td>VARCHAR</td>
</tr>

<tr>
<td><code>conversion</code></td>
<td>BOOLEAN</td>
</tr>
</tbody>
</table>
<code>user_sessions</code> table

<table>
<thead>
<tr>
<th>column</th>
<th>type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>session_id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>created_at</code></td>
<td>DATETIME</td>
</tr>

<tr>
<td><code>user_id</code></td>
<td>INTEGER</td>
</tr>
</tbody>
</table>
Example output:

<table>
<thead>
<tr>
<th><code>user_id</code></th>
<th><code>channel</code></th>
</tr>
</thead>

<tbody>
<tr>
<td>123</td>
<td>facebook</td>
</tr>

<tr>
<td>145</td>
<td>google</td>
</tr>

<tr>
<td>153</td>
<td>facebook</td>
</tr>

<tr>
<td>172</td>
<td>organic</td>
</tr>

<tr>
<td>173</td>
<td>email</td>
</tr>
</tbody>
</table>

A team wants to A/B test multiple different changes through a sign-up funnel.

For example, on a page, a button is currently red and at the top of the page. They want to see if changing a button from red to blue and/or from the top of the page to the bottom of the page will increase click-through.

How would you set up this test?

Your team wants to run an AB test on the color (red or blue) and position (top/bottom of the page) of a button that links to a promotion.

How many different variants of the test should you run to see the effect that a red button has on the bottom of the page, if currently the button is blue and on the top of the page.

Your team wants to run an AB test on the color (red or blue) and position (top/bottom of the page) of a button that links to a promotion.

How many different variants of the test should you run to see the effect that a red button has on the bottom of the page, if currently the button is blue and on the top of the page.

A/B Test Variants

Button AB Test

A/B Testing

Let’s say that your company is running a standard control and variant AB test on a feature to increase conversion rates on the landing page. The PM checks the results and finds a .04 p-value.

How would you assess the validity of the result?

How would you assess the validity of the result?

Experiment Validity

Given a table called <code>user_experiences</code>, write a query to determine the percentage of users that held the title of <code>&#34;Data Analyst&#34;</code> immediately before holding the title <code>&#34;Data Scientist&#34;</code>.

Immediate is defined as the user holding no other titles between the <code>&#34;Data Analyst&#34;</code> and <code>&#34;Data Scientist&#34;</code> roles.

Example:

Input:

<code>user_experiences</code> table

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>position_name</code></td>
<td>VARCHAR</td>
</tr>

<tr>
<td><code>start_date</code></td>
<td>DATETIME</td>
</tr>

<tr>
<td><code>end_date</code></td>
<td>DATETIME</td>
</tr>

<tr>
<td><code>user_id</code></td>
<td>INTEGER</td>
</tr>
</tbody>
</table>
Output:

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>percentage</code></td>
<td>FLOAT</td>
</tr>
</tbody>
</table>

Let’s say we are given a table <code>user_logins</code>.

Using this table, calculate how many users logged in an identical number of times on January 1st, 2022.

For example, it may be the case that on that date three users logged in seven times, four users logged in five times, two users logged in ten times, etc.

Example:

Input:

<code>user_logins</code> table

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>user_id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>login_date</code></td>
<td>DATETIME</td>
</tr>
</tbody>
</table>
Output:

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>number_of_logins</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>number_of_users</code></td>
<td>INTEGER</td>
</tr>
</tbody>
</table>

You have an array of integers, <code>nums</code> of length <code>n</code> spanning <code>0</code> to <code>n</code> with one missing. Write a function <code>missing_number</code> that returns the missing number in the array.

Note: Complexity of \(O(n)\) required.

Example:

Input:

<pre tabindex="0" class="chroma"><code>nums = [0,1,2,4,5] 
missing_number(nums) -&gt; 3
</code></pre>

How would you explain what a p-value is to someone who is not technical?

What does a p-value in a statistical test represent?

What does a p-value in a statistical test represent?

P-value to a Layman I

In a statistical test, how does a low p-value (less than 0.05) influence our decision about the null hypothesis?

In a statistical test, how does a low p-value (less than 0.05) influence our decision about the null hypothesis?

P-value to a Layman II

Explain what a p-value is to someone who is not technical

P-value to a Layman

Statistics

Write a query that creates an attribution rule for each user. If the user visited Facebook or Google at least once then the attribution is labeled as “paid.” Otherwise, the attribution is labeled as “organic.”

Example:

Input:

<code>user_sessions</code> table

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>created_at</code></td>
<td>DATETIME</td>
</tr>

<tr>
<td><code>session_id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>user_id</code></td>
<td>INTEGER</td>
</tr>
</tbody>
</table>
<code>attribution</code> table 

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>session_id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>channel</code></td>
<td>VARCHAR</td>
</tr>
</tbody>
</table>
Output:

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>user_id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>attribute</code></td>
<td>VARCHAR</td>
</tr>
</tbody>
</table>

Write a query to create an attribution rule for each user

Attribution Rules

Let’s say that you work at a bank that wants to build a model to detect fraud on the platform.

The bank wants to implement a text messaging service in addition that will text customers when the model detects a fraudulent transaction in order for the customer to approve or deny the transaction with a text response.

How would we build this model?

Let’s say that you work at a bank that wants to build a model to detect fraud on the platform.

The bank wants to implement a text messaging service that will text customers when the model detects a fraudulent transaction in order for the customer to approve or deny the transaction with a text response.

Which statement is true?

Let's say that you work at a bank that wants to build a model to detect fraud on the platform.

The bank wants to implement a text messaging service that will text customers when the model detects a fraudulent transaction in order for the customer to approve or deny the transaction with a text response.

Which statement is true?

Let’s say that you’re the PM on Google Maps.

<ol>
<li>How would you improve Google Maps?</li>

<li>What metrics would you check to see if your feature improvements are successful?</li>
</ol>

Google Maps Improvement

Product Sense & Metrics

Tell me about a data project that didn’t go the way you expected. What did you set out to do, what surprised you, and how did you handle it?

Let’s say that you work at a B2B SAAS company that’s interested in testing the pricing of different levels of subscriptions.

Your project manager comes to you and asks you to run a two-week-long A/B test to test an increase in pricing.

How would you approach designing this test? How would you determine whether the increase in pricing is a good business decision?

How would you design a two-week A/B test to determine whether a pricing increase is a good business decision

Testing Price Increase

<h1>Binary Tree Validation</h1>

You are given the root of a binary tree. You need to determine if it is a valid binary search tree (BST).

A valid BST is defined as follows:

<ul>
<li>The left subtree of a node contains only nodes with values less than or equal to the node’s value.</li>
<li>The right subtree of a node contains only nodes with values greater than or equal to the node’s value.</li>
<li>Both the left and right subtrees must also be binary search trees.</li>
</ul>

Given the function <code>def is_valid_bst(root: Node) -&gt; bool:</code>, return True if the binary tree is a valid BST. Otherwise, return False.

<h3>Example:</h3>

Input:

<img src="/s3/questions/9664817c-3e12-4a40-9236-8382b13f55a0.png" alt="image"/>Converted Binary Tree.png

Output:

<pre tabindex="0" class="chroma"><code>def is_valid_bst(Node(3)) -&gt; True
</code></pre>

Given the root node, verify if a binary search tree is valid or not.

Binary Tree Validation

Write a Python function called <code>group_anagrams</code> that takes a list of strings as input and returns a list of lists, where each inner list contains a group of anagrams that were present in the input list.

An anagram is a word or phrase formed by rearranging the letters of a different word or phrase, using all the original letters exactly once.

Note: Each of the inner lists should be ordered alphabetically and contain no duplicate values

<h3>Example:</h3>

Input:

<pre tabindex="0" class="chroma"><code>words = [&#34;eat&#34;, &#34;tea&#34;, &#34;tan&#34;, &#34;ate&#34;, &#34;nat&#34;, &#34;bat&#34;]
</code></pre>

Output:

<pre tabindex="0" class="chroma"><code>[ 
 [&#34;nat&#34;,&#34;tan&#34;],
 [&#34;ate&#34;,&#34;eat&#34;, &#34;tea&#34;],
 [&#34;bat&#34;]
]
</code></pre>

Explanation:

In the example, the words “eat”, “tea”, and “ate” are anagrams of each other, as are the words “tan” and “nat”. The word “bat” does not have any anagrams in the input list, and therefore is listed on its own. The function then groups the anagrams together and returns a list of lists representing these groups.

Each of the lists representing a group of anagrams is sorted in alphabetical order.

Identify the groups of anagrams in a list of words

Groups of Anagrams

Write a Python function called <code>longest_increasing_subsequence</code> that takes a list of integers as input and returns as output the length of the longest increasing subsequence in the list.

Note: An increasing subsequence is a subset of a list where the elements are in strictly increasing order. The subsequence does not have to be contiguous or unique.

<h3>Example:</h3>

Input:

<pre tabindex="0" class="chroma"><code>nums = [10, 9, 2, 5, 3, 3, 7, 101, 18]
</code></pre>

Output:

<pre tabindex="0" class="chroma"><code>4
</code></pre>

Explanation:

The longest increasing subsequence is <code>[2, 3, 7, 101]</code>, for a total length of 4.

Find the longest increasing subsequence in a list of integers.

Longest Increasing Subsequence

We want to build a model to predict housing prices in the city of Seattle. We’ve scraped 100K sold listings over the past three years but found that around 20% of the listings are missing square footage data.

How do we deal with the missing data to construct our model?

We want to build a model to predict housing prices in the city of Seattle. We’ve scraped 100K sold listings over the past three years, but have discovered that around 20% of the listings are missing square footage data.

How would you approach dealing with this missing data, in order to construct the most useful predictive model possible?

We want to build a model to predict housing prices in the city of Seattle. We’ve scraped 100K sold listings over the past three years, but have discovered that around 20% of the listings are missing square footage data.

How would you approach dealing with this missing data, in order to construct the most useful predictive model possible?



Given a string <code>str</code> of any length, write an algorithm <code>max_repeating</code> to return which character has the longest string of continuous repetition.

If two characters are tied for most continuous repetition, return the character whose longest continuous repetition occurs earliest in <code>str</code>.

Example 1:

Input:

<pre tabindex="0" class="chroma"><code>str = &#39;aabbaaccbbbaa&#39;
def max_repeating(str) -&gt; b
</code></pre>

Example 2:

Input:

<pre tabindex="0" class="chroma"><code>str = &#39;adccccbbbbc&#39;
def max_repeating(str) -&gt; c
</code></pre>

Write an algorithm to return the character with the longest continuous repetition, breaking ties by earliest occurrence

Most Repetition

Given an array and a target integer, write a function <code>sum_pair_indices</code> that returns the indices of two integers in the array that add up to the target integer. If not found, just return an empty list.

Note: Can you do it on \(O(n)\) time?

Note: Even though there could be many solutions, only one needs to be returned.

Example 1:

Input:

<pre tabindex="0" class="chroma"><code>array = [1 2 3 4] 
target = 5 
</code></pre>

Output:

<pre tabindex="0" class="chroma"><code>def sum_pair_indices(array, target) -&gt; [0 3] or [1 2]
</code></pre>

Example 2:

Input:

<pre tabindex="0" class="chroma"><code>array = [3]
target = 6 
</code></pre>

Output:

<pre tabindex="0" class="chroma"><code>Do NOT return [0 0] as you can&#39;t use an index twice.
</code></pre>

In the context of hypothesis testing, what are type I errors (type one errors) and type II errors (type two errors)? What is the difference between the two?

Bonus: Describe the probability of making each type of error mathematically.

What is the difference between type I and type II errors?

Type I and II Errors

Let’s say you’re a data scientist at LinkedIn where you’re working on a product that sends qualified job candidates to companies. The team has launched a new feature that allows candidates to message hiring managers at companies directly during the interview process to get updates on their status.

Due to engineering constraints, the company can’t AB test the feature before launching it.

How would you analyze how the feature is performing?

How would you analyze the performance of a new LinkedIn feature that lets candidates message hiring managers during the interview process

Recruiting Leads

You’re a data scientist at YouTube. Leadership wants to revisit how the platform measures success, not for a specific new feature, but for the product as a whole given its current state.

How would you define and measure success for YouTube? Identify your primary and secondary metrics, explain your rationale for each, and discuss how you would choose the right thresholds for any engagement-based metrics you propose.

The role of A/B testing in measuring the success rate of an analytics experiment

Success Measurement

In data science, there exists the concept of stemming, which is the heuristic of chopping off the end of a word to clean and bucket it into an easier feature set. 

Given a dictionary consisting of many roots and a sentence, write a function <code>replace_words</code> to stem all the words in the sentence with the root forming it. If a word has many roots that can form it, replace it with the root with the shortest length.

Example:

Input:

<pre tabindex="0" class="chroma"><code>
roots = [&#34;cat&#34;, &#34;bat&#34;, &#34;rat&#34;]
sentence = &#34;the cattle was rattled by the battery&#34;
</code></pre>

Output:

<pre tabindex="0" class="chroma"><code> &#34;the cat was rat by the bat&#34;
</code></pre>

What’s the difference between Lasso and Ridge Regression?

Under what circumstances is it recommended to use Lasso regression over Ridge regression?

Under what circumstances is it recommended to use Lasso regression over Ridge regression?

Imagine you are asked to build a machine learning model to decide new loan approvals for a
financial firm. You ask the data department in the company for a subset of data to get started
working on the problem. The data includes different features about applicants such as age,
occupation, zip code, height, number of children, favorite color, etc. You decide to build
multiple machine learning models to test out different ideas before settling on the best one.

How would you explain the bias-variance tradeoff with regards to
building and choosing a model to use?

Bias vs. Variance Tradeoff

What are the assumptions of linear regression?

Which assumption of the residuals of the standard linear regression model can not be overcome by increasing the sample size?

Regression assumptions

What are the assumptions of linear regression?

Assumptions of Linear Regression

You work for an insurance SAAS company that sells leads to insurance agents.

The VP of sales at your company asks you to dig into whether more leads delivered to the agents provides more value and retains customers for longer periods of time. They point to a graph that shows agents at month three receiving more than 3x the leads than agents in month one as proof.

What is potentially flawed with the VP’s approach?

What is potentially flawed with the VP's approach?

Insurance Leads

Business Case

You’re given a table called <code>annual_payments</code> for an annually billed B2B SAAS subscription product.

<code>annual_payments</code> table

<table>
<thead>
<tr>
<th>Column</th>
<th>Type</th>
</tr>
</thead>

<tbody>
<tr>
<td><code>id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>status</code></td>
<td>VARCHAR</td>
</tr>

<tr>
<td><code>user_id</code></td>
<td>INTEGER</td>
</tr>

<tr>
<td><code>created_at</code></td>
<td>DATETIME</td>
</tr>

<tr>
<td><code>amount</code></td>
<td>FLOAT</td>
</tr>

<tr>
<td><code>amount_refunded</code></td>
<td>FLOAT</td>
</tr>

<tr>
<td><code>product</code></td>
<td>VARCHAR</td>
</tr>

<tr>
<td><code>last_updated</code></td>
<td>DATETIME</td>
</tr>
</tbody>
</table>
Users pay for the three different products: <code>&#39;PDF Editor&#39;</code>, <code>&#39;Cloud Storage&#39;</code>, and <code>&#39;Mobile CRM&#39;</code>.

How would you formulate a query to calculate the average annual retention, for each subsequent year, at the end of the year?

Retention rate is calculated as a percentage of active subscriptions at the end of the year \(T\) in the active subscriptions at the end of the previous year \((T-1)\).

Example 1:

User 1 bought a subscript to <code>PDF editor</code> in 2019 for the first time and renewed their subscription in 2020. In 2021, they canceled their subscription.
Retention rates should be calculated as:

<table>
<thead>
<tr>
<th>2019</th>
<th>2020</th>
<th>2021</th>
</tr>
</thead>

<tbody>
<tr>
<td>0.00</td>
<td>1.00</td>
<td>0.00</td>
</tr>
</tbody>
</table>
Example 2:

User 2 bought a subscription to <code>&#39;PDF Editor&#39;</code> in 2019 for the first time, renewed their subscription in 2020, and canceled a few days later in 2020.
Retention rates should be calculated as:

<table>
<thead>
<tr>
<th>2019</th>
<th>2020</th>
<th>2021</th>
</tr>
</thead>

<tbody>
<tr>
<td>0.00</td>
<td>0.00</td>
<td>0.00</td>
</tr>
</tbody>
</table>
Notes:

<ol>
<li>The status column may contain the values <code>&#39;paid&#39;</code>, <code>&#39;refunded&#39;</code>, or <code>&#39;failed&#39;</code>.</li>
<li>A <code>&#39;failed&#39;</code> payment equates to a user canceling their subscription on the date the next payment is due.</li>
<li>If a user refunds a successful payment, then the row gets updated, and the status becomes <code>&#39;refunded&#39;</code>. The date of refund is recorded in <code>last_updated</code>.</li>
<li>When a refund occurs in year \(T\), the subscription is canceled. Users may purchase the same product again in the subsequent years, and it is considered a new purchase.</li>
</ol>

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