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8
Subject
Statistics
Date
Jan 9, 2024
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Data C8, Final Exam
Summer 2023
Name:
Email:
@berkeley.edu
Student ID:
Name of the student to your left:
Name of the student to your right:
Instructions:
Do not
open the examination until instructed to do so.
This exam consists of
80 points
spread out over
4 questions
on
14 pages
and must be
completed in the
110 minute
time period on August 11, 2023, from 10:10 AM to 12:00
PM unless you have pre-approved accommodations otherwise.
Note that some questions have circular bubbles to select a choice. This means that you
should only
select one choice
. Other questions have boxes. This means you should
select
all that apply
. Please shade in the box/circle to mark your answer.
There is space to write your student ID number (SID) in the upper right-hand corner of
each page of the exam.
Make sure to write your SID on each page
to ensure that your
exam is graded.
Honor Code [1 pt]:
As a member of the UC Berkeley community, I act with honesty, integrity, and respect
for others.
I am the person whose name is on the exam, and I completed this exam in
accordance with the Honor Code.
Signature:
1
Data C8
Final Exam, Page 2 of 16
SID:
1
Barbenheimer Returns [18 Points]
Rotten Tomatoes, a movie review website, is measuring which of the two movies – Oppenheimer
or Barbie – has higher reviews among Berkeley students. They believe that Berkeley students will
give higher reviews to the Oppenheimer movie.
Researchers at Rotten Tomatoes randomly sample
1000
Berkeley students and show
each
student
both
movies under identical viewing conditions. Immediately after watching each movie, every
student is asked to rate that movie on an integer scale from
1
(worst) up to, and including
10
(best).
The reviews are collected in a table named
reviews
; shown below are the first few rows.
(a) [2 Pts] Which of the following is a correct
null
hypothesis that Rotten Tomatoes should use
to assess their claim?
Select one
.
⃝
The Oppenheimer movie has
a different distribution of reviews
than the Barbie
movie among the given sample of Berkeley students.
⃝
The Oppenheimer movie has
the same distribution of reviews
as the Barbie movie
among the given sample of Berkeley students.
⃝
The Oppenheimer movie has
a different distribution of reviews
than the Barbie
movie among Berkeley students.
⃝
The Oppenheimer movie has the same distribution of reviews as the Barbie
movie among Berkeley students.
(b) [2 Pts] Please state a clear and complete
alternative
hypothesis that Rotten Tomatoes should
use to assess their claim.
Solution:
The Oppenheimer movie has higher reviews than the Barbie movie among Berkeley stu-
dents
Data C8
Final Exam, Page 3 of 16
SID:
(c) [3 Pts] Rotten Tomatoes uses the
difference of means
as their test statistic. Complete the
function below so that it returns the difference of mean reviews between the two movies.
Larger values of the test statistic should favor the alternative hypothesis.
Note:
Assume that the
reviews
table
argument resembles the
reviews
table above.
Hint
: The
group
function will return a table that is sorted alphabetically based on the values
in the column used for grouping.
def test_statistic(reviews_table):
means_col = ___________________(A)________________________
return ________________________(B)________________________
(i) Fill in the blank
(A)
Solution:
reviews_table.group(0, np.mean).column(1)
(ii) Which of the following options is most appropriate for blank
(B)
⃝
means_col.item(0) - means_col.item(1)
⃝
means_col.item(1) - means_col.item(0)
(d) [3 Pts] Which of the following may be used to create simulations under the null hypothesis?
Select all that apply.
□
Shuffle the values of only the
movie
column.
□
Shuffle the values of only the
review
column.
□
Shuffle the values of the
movie
column, then shuffle the values of the
review
column.
□
Randomly sample all of the rows of the
reviews
table
with replacement
.
□
Randomly sample all of the rows of the
reviews
table
without replacement
.
□
None of the above.
(e) [2 Pts] Suppose we simulate
10
,
000
values of the test statistic under the null hypothesis.
Which of the following will our distribution of simulated test statistics most closely resemble?
⃝
Graph 1
⃝
Graph 2
⃝
Graph 3
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Data C8
Final Exam, Page 4 of 16
SID:
(f) [3 Pts] You obtain a
p
-value of
0
.
37
from your experiment above. Which of the following
statements are true?
Select all that apply.
Note
: Recall that larger values of your test statistic should favor the alternative hypothesis.
□
Your observed test statistic lies at the
63
rd
percentile of the distribution of test
statistics simulated under the null hypothesis.
□
37%
of the test statistics simulated under the null hypothesis were as, or less ex-
treme than the observed test statistic.
□
The Barbie movie has higher reviews than the Oppenheimer movie among Berkeley
students.
□
With a
p
-value cutoff of
5%
, our data are consistent with the null hypothesis.
□
None of the above.
(g) [3 Pts] Which of the following statements are true?
Select all that apply.
□
If Rotten Tomatoes repeats the same experiment, but instead, they sample
10
,
000
Berkeley students, the observed test statistic will more accurately re-
flect whether Oppenheimer is reviewed higher than Barbie among Berkeley
students.
□
If Rotten Tomatoes repeats the same experiment, but instead, they sample
10
,
000
Berkeley students, the distribution of test statistics simulated under
the null hypothesis will have a smaller standard deviation.
□
If Rotten Tomatoes repeats the same experiment, but instead,
they simulate
1000
values of the test statistic under the null hypothesis
, the distribution of these
simulated test statistics will have a larger standard deviation.
□
None of the above.
Data C8
Final Exam, Page 5 of 16
SID:
2
California Loves Transit [22 Points]
You’ve just been hired as a data scientist for the City of San Francisco! Your team is interested
in studying public transportation, so you begin analyzing data from the widely-used BART train
system and the AC Transit bus services during 2022.
Unfortunately, there is so much data from 2022 that it will overwhelm your computer, so instead,
your team gives you a large random sample of
1000
riders in a table called
transport
. Displayed
below are the first few rows.
•
id
(
integer
): identification (id) of the rider.
•
transfer
(
boolean
): whether that particular rider transferred between a BART train and
an AC Transit bus at least once during 2022.
•
fares
(
float
): total amount that particular rider spent on fares in 2022, measured in dollars.
(a) [2 Pts] Given below is the distribution of the
fares
column from the
transport
table.
Which of the following conclusions can you draw from the plot?
Select all that apply
.
□
The distribution of the
fares
col-
umn in
transport
is right-skewed.
□
The distribution of the
fares
column
in
transport
is left-skewed.
□
The median of the
fares
column in
transport
is less than the mean.
□
The median of the
fares
column
in
transport
is
greater than
the
mean.
Data C8
Final Exam, Page 6 of 16
SID:
(b) [2 Pts] Which of the following statements must be true?
Select all that apply
.
□
The distribution of
fare spending
among all riders is approximately normal.
□
The distribution of sample means of fare spending is approximately normal
for large random samples of data.
□
The distribution of sample sums of fare spending is approximately normal for
large random samples of data.
□
The distribution of
sample medians of fare spending
is approximately normal for
large random samples of data.
□
None of the above.
Your team is interested in estimating the proportion of all riders who had transferred between a
BART train and an AC Transit bus at least once. You decide to use your sample of
1000
riders to
estimate this unknown population parameter.
(c) [4 Pts] Fill in the blanks to generate a visualization of
10
,
000
bootstrapped proportions of
riders who transferred between a BART train and an AC Transit bus at least once.
resample_props = make_array()
for i in np.arange(10000):
resamp = ________________________(A)___________________
resamp_prop = ___________________(B)___________________
__________________________(C)__________________________
Table().with_column("resample_props", resample_props).hist()
Fill in the blank
(A)
Solution:
transport.sample()
Fill in the blank
(B)
Solution:
np.mean(resamp.column("transfer"))
Fill in the blank
(C)
Solution:
resample_props = np.append(resample_props, resamp_prop)
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Data C8
Final Exam, Page 7 of 16
SID:
(d) [2 Pts] You find that the mean and standard deviation of your bootstrapped proportions,
resample
props
is
0
.
37
and
0
.
015
, respectively.
Which of the following most closely
resembles the distribution of
resample
props
?
⃝
Graph 1
⃝
Graph 2
⃝
Graph 3
(e) [3 Pts] Write a mathematical expression that evaluates to the probability that the first row in
transport
is included
at least once
in a single bootstrap re-sample of size
1000
.
Please
do not simplify.
Solution:
P(the first row is included at least once in one bootstrap sample)
1
−
P(the first row is not chosen for any of the bootstrap’s
1000
rows)
1
−
P(the first row is not chosen)
1000
1
−
(
999
1000
)
1000
(f) [2 Pts] Fill in the blanks so that
interval
contains the left and right endpoints of a
95%
confidence interval for the proportion of riders in the population who transferred at least once.
Note
: You may used variable names defined from previous sub-parts in your code.
left = ___________________(A)___________________
right = ___________________(B)___________________
interval = make_array(left, right)
Data C8
Final Exam, Page 8 of 16
SID:
Fill in the blank
(A)
Solution:
percentile(2.5, resample_props)
Fill in the blank
(B)
Solution:
percentile(97.5, resample_props)
(g) [3 Pts] Which of the following conclusions can you draw using your
95%
confidence interval
in part (f)
?
Select all that apply
.
□
If someone takes the BART train, there is a
95%
chance that they transfer to an AC
Transit bus.
□
If you make confidence intervals from many large random samples from the
population, you can expect that roughly
95%
of the intervals you create will
contain the true population proportion.
□
There is a
95%
chance that the population’s true transfer proportion is within the
interval generated in
part (f)
.
□
There is a
95%
chance that the sample’s true transfer proportion is within the inter-
val generated in
part (f)
.
□
None of the above.
(h) [4 Pts] Your team has one last request. They want your
95%
confidence interval to be no
wider than
5%
.
Using the maximum standard deviation of a
0
−
1
population, what is the
smallest sample size that satisfies this requirement?
Express your answer as an integer.
Solution:
Recall that the maximum standard deviation of a
0
−
1
population is
0
.
5
.
0
.
05 = 4
∗
0
.
5
√
samplesize
√
samplesize
= 4
∗
0
.
5
0
.
05
√
samplesize
= 4
∗
10
samplesize
= 1600
Data C8
Final Exam, Page 9 of 16
SID:
3
Breaking Batter: Fried Chicken Edition [23 Points]
Walter and Jesse own a fried chicken restaurant, where they track various details about their food
quality. They store this information in a table called
data
; displayed below are the first few rows.
Every row corresponds to a distinct order of fried chicken, and the data was collected randomly.
Assume that larger values on a
1
−
10
scale are considered better (and smaller values worse).
•
chicken
quality
(
float
): quality of the raw chicken (
scale:
[1
.
0
−
10
.
0]
)
•
cooking
temp
(
integer
): cooking temperature of the fried chicken, in degrees Fahrenheit
•
seasoning
amount
(
integer
): amount of seasoning in the fried chicken, in grams
•
resting
time
(
float
): resting time of the fried chicken before serving, in minutes
•
customer
score
(
float
): customer satisfaction rating of fried chicken (
scale:
[1
.
0
−
10
.
0]
)
(a) [2 Pts] Walter calculates a correlation
r
= 0
.
6
between the two variables
customer
score
and
chicken
quality
. Which of the following conclusions can he draw from this corre-
lation?
Select one.
⃝
Fried chicken made from higher quality chicken generally tends to have higher
customer satisfaction scores than fried chicken made from lower quality chicken.
⃝
In the
data
table, the
customer
score
values generally deviate less from their
average than the
chicken
quality
scores deviate from their average.
⃝
Fried chicken made from the highest quality chicken also has the highest customer
satisfaction score.
⃝
The use of better quality chicken in the fried chicken recipe causes higher customer
satisfaction scores.
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Data C8
Final Exam, Page 10 of 16
SID:
(b) [2 Pts] Given the correlation of
0
.
6
between
chicken
quality
and
customer
score
,
mark the following as
True or False
.
(i) The correlation between
chicken
quality
in
standard units
and
customer
score
in
standard units
is
0
.
6
.
⃝
True
⃝
False
(ii) The correlation between
chicken
quality
in
standard units
and
customer
score
in
original units
is
0
.
6
.
⃝
True
⃝
False
Walter wants to predict the
customer
score
from
chicken
quality
. For the following
parts, you may assume that:
• The
chicken
quality
column has a mean of
8
.
4
and a standard deviation of
0
.
7
• The
customer
score
column has a mean of
8
.
6
and a standard deviation of
0
.
5
• The correlation between
chicken
quality
and
customer
score
is
0
.
6
(c) [4 Pts] What are the
slope
and
intercept
of the regression line in
original units
? You do not
need to simplify; you may write your answer as a mathematical expression.
Note:
In your expression for the intercept in
part (ii)
, you may use the word “slope” to rep-
resent the value of the slope in
part (i)
.
(i) Slope:
Solution:
0.6
*
(0.5)/(0.7)
(ii) Intercept:
Solution:
8.6 - (slope
*
8.4)
(d) [2 Pts] The restaurant receives an exceptional shipment of raw chicken. This shipment has
a
chicken
quality
that is
2
standard deviations above the mean. What is the predicted
satisfaction score
in standard units
that customers will give the fried chicken made from this
new shipment?
Please simplify your answer.
Solution:
1.2
y_su = r
*
x_su
y_su = 0.6
*
2 = 1.2
Data C8
Final Exam, Page 11 of 16
SID:
(e) [2 Pts] The first order of fried chicken made from the shipment in
part (d)
was cooked poorly,
leading to a below average
customer
score
. If Walter adds this order to his
data
table
and fits a new regression line on all the orders in
data
, will the slope of the line increase or
decrease as compared to the regression line in
part (c)
?
Select one.
⃝
Increase
⃝
Decrease
⃝
Not enough information
(f) [3 Pts] To verify Walter’s calculations, Jesse uses an optimization approach to find the least
squares line that predicts
customer
score
from
chicken
quality
. Fill in the blanks
so that
parameters
evaluates to an array of the slope and intercept of the least squares line
that minimizes
root mean squared error
.
def rmse(slope, intercept):
y_predicted = ____________________(A)_____________________
return ________________________(B)________________________
parameters = ___________________(C)___________________________
Fill in the blank
(A)
Solution:
slope
*
data.column("chicken_quality") + intercept
Fill in the blank
(B)
Solution:
np.mean((data.column("customer_score") - y_predicted)
**
2)
**
0.5
Fill in the blank
(C)
Solution:
minimize(rmse)
(g) [2 Pts]
[Fill in the Blank]:
The slope and intercept that Jesse finds from his optimization ap-
proach will be
________
Walter’s slope and intercept values from his regression approach.
⃝
greater than
⃝
less than
⃝
equal to
⃝
Not enough information
Data C8
Final Exam, Page 12 of 16
SID:
Walter now attempts to predict
customer
score
from each of the other variables in the
data
table:
cooking
temp
,
seasoning
amount
, and
resting
time
.
Jesse hands him three scatter plots and claims that these are the residual plots from the regression
line that predicts
customer
score
from each of the three variables above.
Plot 1
Plot 2
Plot 3
Do each of the plots indicate that Jesse used the
regression line
to predict
customer
score
? If
you answer
No
, explain in
one sentence
how you know that Jesse did not use the regression line.
Please do not write anything if you answer
Yes
.
(i) [2 Pts]
Plot 1
:
cooking
temp
vs
customer
score
⃝
Yes
⃝
No
Solution:
N/A
(j) [2 Pts]
Plot 2
:
seasoning
amount
vs
customer
score
⃝
Yes
⃝
No
Solution:
Walter should not see association in the residual plot.
(k) [2 Pts]
Plot 3
:
resting
time
vs
customer
score
⃝
Yes
⃝
No
Solution:
Walter should not see residuals centered at the value of -1 – rather they should
be centered at 0.
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Data C8
Final Exam, Page 13 of 16
SID:
4
It’s Always Meme Friday [16 Points]
As you may know, Kevin likes to share memes before the start of lecture, but he is concerned that
students don’t appreciate them. He presents
200
randomly selected lecture memes to all Data 8
students in hopes of understanding whether they like each meme or not. He records the data in a
table called
meme
data
. Each row represents a meme, and the columns are as follows:
•
category
(
string
): the category of the meme, which is either an “image” or a “video”.
•
insta
num
(
integer
): the number of times that meme has been shared on Instagram.
•
time
(
integer
): the duration of the meme, in seconds. Images will have a
time
value of
0
.
•
nontext
percentage
(
float
): the percentage of the meme that is non-textual content
(
scale:
[0
.
0
−
100
.
0]
).
•
rating
(
float
): the percentage of Data 8 students who liked the meme (
scale:
[0
.
0
−
100
.
0]
).
(a) [4 Pts] Choose which single technique is the most appropriate for answering each scenario.
Select one answer choice for each subpart.
Note
: Please select the
“None of the above”
option if the scenario cannot be answered from
the
meme
data
table alone.
(i) Kevin wants to estimate the mean
rating
for all his memes among all Data 8 students.
⃝
Linear Regression
⃝
Bootstrapping
⃝
A/B Testing
⃝
Classification
⃝
None of the above
(ii) Kevin wants to create a model that predicts the
rating
of a meme from the number of
times it has been shared on Instagram.
⃝
Linear Regression
⃝
Bootstrapping
⃝
A/B Testing
⃝
Classification
⃝
None of the above
(iii) Kevin wants to use the
time
column to predict what
category
a meme belongs to.
⃝
Linear Regression
⃝
Bootstrapping
⃝
A/B Testing
⃝
Classification
⃝
None of the above
(iv) Kevin wants to use the number of times a given meme has been shared on Instagram to
predict whether or not some particular Data 8 student will like the meme.
⃝
Linear Regression
⃝
Bootstrapping
⃝
A/B Testing
⃝
Classification
⃝
None of the above
Data C8
Final Exam, Page 14 of 16
SID:
Kevin is interested in building a classification model that uses the numerical features in the
meme
data
table to predict whether a meme will be “popular” or not. Here, a “popular” meme is one that is
liked by more than
50%
of the Data 8 students.
(b) [2 Pts] Please complete the code below so that
meme
popular
is a copy of
meme
data
with an additional column called “
popular
”.
The “
popular
” column should include
boolean values that indicate whether a meme is popular (
True
) or not (
False
).
pop_arr = ____________________(A)____________________
meme_popular = ______________________(B)______________________
Fill in the blank
(A)
Solution:
meme_data.column('rating') > 50
Fill in the blank
(B)
Solution:
meme_data.with_column('popular', pop_arr)
(c) [2 Pts] Kevin converts the
time
and
nontext
percentage
columns to
standard units
and creates two
k
-NN classifiers, each with a different value of
k
:
k
=3
and
k
=9
. Which of
the following plots corresponds to the
3
-NN classifier?
⃝
Visualization A
⃝
Visualization B
Data C8
Final Exam, Page 15 of 16
SID:
(d) [4 Pts] Kevin divides his data into a training and testing data set.
After training a
1
-NN
classifier, he notices that only 10% of the memes in the training data are popular, compared to
50% of memes in the testing data. He finds that this imbalance is due to an error in his code.
After correcting the error and re-distributing the data to restore the balance of popular memes,
Kevin re-trains a
1
-NN classifier. How would you expect the training and testing performance
to change after re-balancing the data?
Training Accuracy
⃝
Increases
⃝
Remains the same
⃝
Decreases
Testing Accuracy
⃝
Increases
⃝
Remains the same
⃝
Decreases
(e) [4 Pts] Before using Kevin’s classifier, a GSI guesses whether a meme from the test set is
popular among Data 8 students. The GSI is accurate 75% of the time. For memes that the GSI
predicts correctly, Kevin’s model’s accuracy is 82%; otherwise, Kevin’s model’s accuracy is
45%. Suppose we randomly sample a meme from the test set and Kevin’s model predicts its
class correctly. What is the probability that the GSI’s prediction is right?
Write your answer
as a mathematical expression.
Solution:
0
.
75
∗
0
.
82
0
.
75
∗
0
.
82 + 0
.
25
∗
0
.
45
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Data C8
Final Exam, Page 16 of 16
SID:
5
Congratulations [0 Pts]
Congratulations! You have completed the Final Exam.
•
Make sure that you have written your student ID number on
each page
of the exam.
You may lose points on pages where you have not done so.
• Also ensure that you have
signed the Honor Code
on the cover page of the exam for 1 point.
[Optional, 0 pts] Draw a picture (or graph) describing your experience in Data 8.
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