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Instructor(s): Justin Eldridge

This exam was administered in-person. The exam was closed-notes, except students were allowed to bring a single two-sided cheat sheet. No calculators were allowed. Students had 80 minutes to take this exam.

Welcome to the Midterm Exam for DSC 80 in Fall 2022!

Throughout this exam, we will be using to the following DataFrame tasks.

Each row of this DataFrame is a recorded task. The columns are as follows:

• 'category': The category that the task falls into. E.g., 'work'.
• 'completed': Whether the task has been completed (True) or not (False).
• 'minutes': The number of minutes that the task took to complete. Is always missing when the task was not completed, but can still be missing even if the task was completed.
• 'urgency': A rating of the task’s importance on a scale from 1 to 3, with 3 being the most important. Some values are missing.
• 'client': Some of the tasks were performed as part of a consulting gig. If so, the client that the task was performed for is listed in this column. Missing values in this column are represented by the string 'n/a'. You can assume that the client is missing if and only if the category is not consulting.

The table’s index contains unique task identifiers, which are simply numbers. E.g., the first row’s task identifier is simply 0.

To find out more about the data set, you have run tasks.info() and performed .value_counts() on most columns. Here are the results:

>>> tasks.info()
<class 'pandas.core.frame.DataFrame'>
Int64Index: 9836 entries, 0 to 9835
Data columns (total 5 columns):
 #   Column     Non-Null Count  Dtype  
---  ------     --------------  -----  
 0   category   9836 non-null   object 
 1   completed  9836 non-null   bool   
 2   minutes    6442 non-null   float64
 3   urgency    8864 non-null   float64
 4   client     1145 non-null   object 
dtypes: bool(1), float64(2), object(2)
memory usage: 393.8+ KB

>>> tasks["category"].value_counts(normalize=True)
work            0.562424
relationship    0.136539
consulting      0.116409
hobbies         0.091094
health          0.046157
finance         0.036702
learning        0.010675
Name: category, dtype: float64

>>> tasks["completed"].value_counts(normalize=True)
True     0.654941
False    0.345059
Name: completed, dtype: float64

>>> tasks["urgency"].value_counts(normalize=True)
2.0    0.502595
1.0    0.392825
3.0    0.104580
Name: urgency, dtype: float64

>>> tasks["client"].value_counts(normalize=True)
The Government of Luxembourg    0.224454
San Diego Financial Analysts    0.200873
ABC LLC                         0.196507
SDUSD                           0.191266
NASA                            0.186900
Name: client, dtype: float64

Problem 1

What kind of data is in the 'minutes' column in the tasks DataFrame?

• Quantitative

• Ordinal

• Nominal

Problem 2

What kind of data is in the 'urgency' column in the tasks DataFrame?

• Quantitative

• Ordinal

• Nominal

Problem 3

Define a 'personal' task to be one whose category is not 'work' or 'consulting'.

Write a piece of code which adds a new column named 'personal' to tasks. An entry of the column should be True if the task is a 'personal' task, and False otherwise.

Problem 4

Write one line of code which returns a Series containing the median number of minutes taken for tasks in each category.

Problem 5

Fill in the code below so that it computes a Series containing, for each category, the proportion of completed tasks which took more than 30 minutes to complete. That is, out of all tasks which were completed in a given category, what percentage took more than 30 minutes?

def proportion_more_than_30(df):
    ____
    

result = (
    tasks.groupby('category')[['completed', 'minutes']]
    .____(proportion_more_than_30)
)

Problem 6

Suppose the date of each task has been collected in a Series named , shown below:

>>> dates
7377   2019-08-31
7378   2019-08-31
7379   2019-09-01
7380   2019-09-02
7381   2019-09-02
            ...    
9831   2022-10-28
9832   2022-10-29
9833   2022-10-29
9834   2022-10-30
9835   2022-10-30
Length: 2459, dtype: datetime64[ns]

You may assume that the index of dates contains the task identifier, and that these identifiers correspond to the same identifiers used in the index of tasks. Note that dates have only been collected for the last 2459 tasks.

What will happen if we try to run the following line of code?

tasks['date'] = dates

• A new row will be created with index “date”.

• A new column will be created with missing values in the 'date' column for tasks that are not in dates.

• An exception will be raised because some tasks are missing (the size of dates is not the same as the size of tasks.)

• An exception will be raised because tasks does not have a column named 'date'.

Problem 7

The code below creates a pivot table.

pt = tasks.pivot_table(
    index='urgency', columns='category', values='completed', aggfunc='sum'
)

Which of the below snippets of code will produce the same result as pt.loc[3.0, 'consulting']? Select all that apply.

Snippet 1:

tasks[
    (tasks['category'] == 'consulting')
    &
    (tasks['urgency'] == 3.0)
]['completed'].sum()

Snippet 2:

tasks[tasks['urgency'] == 3]
.groupby('category')['completed']
.sum().loc['consulting']

Snippet 3:

tasks.groupby('urgency')['completed']
.sum().loc[3.0, 'consulting']

Snippet 4:

tasks.groupby(['urgency', 'category'])['completed']
.sum().loc[(3.0, 'consulting')]

Snippet 5

tasks.groupby('completed').sum().loc[(3.0, 'consulting')]

• Snippet 1

• Snippet 2

• Snippet 3

• Snippet 4

• Snippet 5

Problem 8

Write a line of code which creates a pivot table showing the number of tasks of each urgency level for each client. The index should contain the urgency levels, and the columns should contain the client names.

tasks.pivot_table(
    columns='client', index='urgency', values='category', aggfunc='count'
)

Problem 9

Suppose that in addition to tasks, you have the DataFrame named clients, shown below:

The index of the DataFrame contains the names of clients that have been consulted for, the 'rate' column contains the pay rate (in dollars per hour), and the 'active' column says whether the client is actively being consulted for. Note that the clients which appear in 'clients' are not exactly the same as the clients that appear in tasks["client"].value_counts(). That is, there is a client in tasks["client"] which is not in clients, and a client that is in clients that does not appear in tasks.

Fill in the code below so that it produces a DataFrame which has all of the columns that appear in tasks, but with two additional columns, rate and activity, listing the pay rate for each task and whether the client being consulted for is still active. The number of rows in your resulting DataFrame should be equal to the number of rows in tasks for which the value in 'client' is in clients.

tasks.merge(
    clients,
    how=___,
    ___,
    ___
)

Problem 10

You observe that tasks in the 'consulting' category seem to be completed at a higher rate than tasks in the 'work' category. Could this be due to chance? To check, you’ll run a permutation test. Your hypotheses are: - Null: The true probability of completing a 'work' task is the same as the probability of completing a 'consulting' task. - Alternative: The true probability of completing a 'consulting' is higher than the probability of completing a 'work' task.

In the box below, write code which performs a permutation test and computes a p-value. The choice of an appropriate test statistic is left to you. You should check 10,000 permutations.

Note: you do not need to do anything special to account for missing values.

df = tasks[tasks['category'].isin(['work', 'consulting'])]
df = df[['category', 'completed']]

def test_stat(df):
    """Difference in proportion completed: consulting - work."""
    props = df.groupby('category')['completed'].mean()
    return props['consulting'] - props['work']

observed_stat = test_stat(df)

simulated = []
for i in range(10000):
    df['completed'] = np.random.permutation(df['completed'])
    simulated.append(test_stat(df))
    
p_value = (np.array(simulated) >= observed_stat).mean()

Problem 11

Suppose you wish to perform a permutation test to check the following hypotheses:

• Null: The urgencies of completed tasks are drawn from the same distribution as the urgencies of uncompleted tasks.
• Alternative: The distribution of urgencies for completed tasks is different from the distribution of urgencies for uncompleted tasks.

What is the best test statistic for this test?

• the Kolmogorov-Smirnov statistic

• the Total Variation Distance

• the signed difference between the mean urgency of completed tasks and the mean urgency of uncompleted tasks

• the absolute difference between the mean urgency of completed tasks and the mean urgency of uncompleted tasks

Problem 12

Suppose you are performing a permutation test to check the following hypotheses:

• Null: The time required to complete tasks in the 'hobbies' category is drawn from the same distribution as the time required to complete tasks in the 'finance' category.
• Alternative: The time required to complete tasks in the 'hobbies' category is drawn from a different distribution as the time required to complete tasks in the 'finance' category.

When you plot a histogram for the distribution of times taken to complete tasks in each categories, you see the below:

What is the best test statistic for this test?

• the Kolmogorov-Smirnov statistic

• the Total Variation Distance

• the signed difference between the mean time taken for 'finance' tasks, minus the mean time taken for 'hobbies' tasks.

• the maximum time taken for 'finance' tasks, minus the maximum time taken for 'hobbies' tasks

Problem 13

What is the most likely type of missingness for the missing values in the ‘client’ column of tasks?

Hint: look at the description of the ‘client’ column at the beginning of the exam for information on when missing values appear.

• Missing by Design

• Missing Completely at Random

• Not Missing at Random

• Missing at Random

This question is designed to have a single most likely answer. However, you can optionally provide a justification for your answer below. If your answer above is correct, you will get full credit even if you do not provide justification. However, if your answer above is wrong, you may receive some credit if you provide justification and it is reasonable.

Problem 14

Suppose it is known that the person who collected this data set often neglected to enter an urgency for tasks that they knew would take less than 5 minutes, and that this factor was the most important in the missingness in the ‘urgency’ column.

What is the most likely type of missingness for the missing values in the ‘urgency’ column?

• Missing by Design

• Missing Completely at Random

• Not Missing at Random

• Missing at Random

This question is designed to have a single most likely answer. However, you can optionally provide a justification for your answer below. If your answer above is correct, you will get full credit even if you do not provide justification. However, if your answer above is wrong, you may receive some credit if you provide justification and it is reasonable.

Problem 15

Suppose you wish to run a permutation test to determine whether the missingness in the 'urgency' column is Missing at Random because of an association with values in the 'category' column. Which test statistic should be used?

• The TVD between the distribution of categories in tasks where the urgency is missing, and the distribution of categories in tasks where the urgency is not missing.

• The Kolmogorov-Smirnov statistic between the distribution of categories in tasks where the urgency is missing, and the distribution of categories in tasks where the urgency is not missing.

• The TVD between the distribution of urgencies in tasks where the category is missing, and the distribution of urgencies in tasks where the category is not missing.

• The Kolmogorov-Smirnov statistic between the distribution of urgencies in tasks where the category is missing, and the distribution of urgencies in tasks where the category is not missing.

Problem 16

Suppose it is determined that the missingness in the 'urgency' column is Missing at Random due to an association with the 'category'column, and that tasks in the 'work' category are least likely to be missing an urgency. You can assume that tasks in the 'work' category have higher-than-average urgencies.

Suppose the missing urgencies are imputed by randomly sampling from the observed urgencies. If the mean urgency of tasks in the 'hobbies' category is computed, what is likely to be true?

• It will be unbiased.

• It will be biased high (higher than the true average).

• It will be biased low (lower than the true average).

Problem 17

Provide an 1 or 2 sentence argument for why the missingness in the 'minutes' column may be Not Missing at Random.

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