Answer: The confounding factor that Dasha hasn’t considered is the prevalence of EVs. There are a lot more EVs now than ever before, so even if EVs are made to the same standards as gasoline cars or better, we’d expect to see more repairs of EVs now.

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 68%.

Answer: Histogram

"Range" is a numerical (i.e. quantitative) variable, and we use histograms to visualize the distribution of numerical variables.

• A bar chart couldn’t work here. Bar charts can show the distribution of a categorical variable, but "Range" is not categorical.
• A scatter plot visualizes the relationship between two numerical variables, but we are only dealing with one numerical variable here ("Range").
• Similarly, a line plot visualizes the relationship between two numerical variables, but we only have one here.

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 63%.

Answer:

• A bar chart that shows the average "Range" for each "Brand"
• An overlaid histogram showing the distribution of "Range" for each "Brand"

Let’s look at each option more closely.

• Option 1: A bar chart showing the distribution of "Brand" would only show us how many cars of each "Brand" there are. It would not tell us anything about the average "Range" of each "Brand".

• Option 2: A bar chart showing the average range for each "Brand" would help us directly visualize how the average range of each "Brand" compares to one another.

• Option 3: An overlaid histogram, although perhaps a bit messy, would also give us a general idea of the average range of each "Brand" by giving us the distribution of the "Range" of each brand. In the scenario mentioned in the question, we’d expect to see that the Tesla distribution is further right than the BMW distribution.

• Option 4: A scatter plot of "TopSpeed" against "Range" would only illustrate the relationship between "TopSpeed" and "Range", but would contain no information about the "Brand" of each EV.

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Difficulty: ⭐️

The average score on this problem was 91%.

Answer: No

"Seats" is a numerical variable, since it makes sense to do arithmetic with the values. For instance, we can find the average number of "Seats" that a group of cars has. Gabriel’s argument could apply to any numerical variable; just because we can place numerical variables into “bins” doesn’t make them categorical.

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 51%.

Answer:

• (i): Series
• (ii): 32

The .apply method applies a function on every element of a Series. Here, evs.get("Brand").apply(max) applies the max function on every element of the "Brand" column of evs, producing a new Series with the same length as evs.

While not necessary to answer the question, if s is a string, then max(s) evaluates to the single character in s that is last in the alphabet. For instance, max("zebra") evaluates to "z". As such, evs.get("Brand").apply(max) is a Series of 32 elements, each of which is a single character, corresponding to the latest character in the alphabet for each entry in evs.get("Brand").

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 65%.

Answer:

• (i): string
• (ii): 5

The .max() method will find the “largest” element in the Series it is called in, which in this case is evs.get("Brand"). The way that strings are ordered is alphabetically, so evs.get("Brand").max() will be the last value of "Brand" alphabetically. Since we were told that the only values in the "Brand" column are "Tesla", "BMW", "Audi", and "Nissan", the “maximum” is "Tesla", which has a length of 5.

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 54%.

Answer: 32

counts is a DataFrame with one row per "Brand", since we grouped by "Brand". Since we used the .count() aggregation method, the columns in counts will all be the same – they will all contain the number of rows in evs for each "Brand" (i.e. they will all contain the distribution of "Brand"). If we sum up the values in any one of the columns in counts, then, the result will be the total number of rows in evs, which we know to be 32. Thus, counts.get("Range").sum() is 32.

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 56%.

Answer: "Tesla"

Since we grouped by "Brand" to create counts, the index of counts will be "Brand", sorted alphabetically (this sorting happens automatically when grouping). This means that counts.index will be the array-like sequence ["Audi", "BMW", "Nissan", "Tesla"], and counts.index[3] is "Tesla".

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Difficulty: ⭐️⭐️⭐️⭐️

The average score on this problem was 33%.

Answer: .groupby("Brand")

When we group by a column, the resulting DataFrame contains one row for every unique value in that column. The question specified that we wanted some information per "Brand", which implies that grouping by "Brand" is necessary.

After grouping, we need to use an aggregation method. Here, we wanted the resulting DataFrame to be such that the "Seats" column contained the average number of "Seats" per "Brand"; this is accomplished by using .mean(), which is already done for us.

Note: With the provided solution, the resulting DataFrame also has other columns. For instance, it has a "Range" column that contains the average "Range" for each "Brand". That’s fine, since we were told that the resulting DataFrame may have other columns in it as well. If we wanted to ensure that the only column in the resulting DataFrame was "Seats", we could have used .get(["Brand", "Seats"]) before grouping, though this was not necessary.

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Difficulty: ⭐️⭐️

The average score on this problem was 76%.

Answer: [evs.get("Brand") == "Audi"].get("TopSpeed")

There are two parts to this problem:

1. Querying, to make sure that we only keep the rows corresponding to Audis. This is accomplished by:

   • Using evs.get("Brand") == "Audi" to create a Boolean Series, with Trues for the rows we want to keep and Falses for the other rows.
   • Using Boolean indexing to keep only the rows in which the aforementioned Series is True. This is accomplished by evs[evs.get("Brand") == "Audi"] (though the evs part at the front was already provided).

2. Accessing the "TopSpeed" column. This is accomplished by using .get("TopSpeed").

Then, evs[evs.get("Brand") == "Audi"].get("TopSpeed") is a Series contaning the "TopSpeed"s of all Audis, and mean of this Series is the result we’re looking for. The call to .mean() was already provided for us.

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Difficulty: ⭐️⭐️

The average score on this problem was 77%.

Answer: .get("TopSpeed").apply(np.log)

The .apply method is used to apply a function on every element of a Series. The relevant Series here is the column containing the "TopSpeed" of each EV, i.e. evs.get("TopSpeed") (the evs part was already provided to us).

After we get that Series, we need to use the function np.log on every element of it. This is accomplished by using .apply(np.log). Putting our steps so far together, we have evs.get("TopSpeed").apply(np.log), which is a Series containing the natural logarithm of the "TopSpeed" of all EVs in evs.

The number we were asked for was the average of the natural logarithm of the "TopSpeed" of all EVs in evs; all we need to do now is use the .mean() method at the end, which was already done for us.

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 61%.

Answer: Audi

Let’s compute the number of EVs that are either SUVs or sedans for each non-Nissan Brand. (To do this, we’ll look at the right-most two columns in the DataFrame provided.)

• Number of Teslas that are SUVs or sedans: 4 + 3 = 7
• Number of BMWs that are SUVs or sedans: 1 + 1 = 2
• Number of Audis that are SUVs or sedans: 8 + 1 = 9

Since Audi is the "Brand" with the most total SUVs and sedans, it is the most likely "Brand" to be selected.

Note: You could compute conditional probabilities for each brand, if you’d like, by dividing the counts above by 18 (the total number of SUVs and sedans). For instance, P(\text{EV is a BMW given that EV is an SUV or sedan}) = \frac{2}{18}. The "Brand" with the highest count (Audi, with 9 SUVs or sedans) is also the "Brand" with the highest conditional probability of being selected given that the selected car is an SUV or sedan (Audi, with \frac{9}{18}).

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Difficulty: ⭐️⭐️

The average score on this problem was 88%.

Answer: 92

In the first selection, the probability of selecting a BMW is \frac{1+1+1}{24} = \frac{3}{24} (3 is the total number of EVs that are BMW, and 24 is the total number of non-Nissan EVs as given by the question).

In the second selection, since we select without replacement, there are only 23 EVs we can select from. Given that in the first selection we already selected 1 BMW, there are only 2 BMWs left among the 23 EVs left. Thus, the probability of getting a BMW in the second selection is \frac{2}{23}.

Putting this all together, the probability that both selections are BMWs is

\frac{3}{24}\cdot\frac{2}{23} = \frac{6}{24} \cdot \frac{1}{23} =\frac{1}{4} \cdot \frac{1}{23} = \frac{1}{92}

So, k = 92.

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 67%.

Answer: \frac{4}{13}

The question is asking for the proportion of SUVs that are made by Tesla.

We first need to find the number of SUVs in the DataFrame provided, which is 4 + 1 + 8 = 13. Of those 13 SUVs, 4 are made by Tesla. Thus, the proportion of SUVs made by Tesla is \frac{4}{13}, so the probability that a randomly selected SUV is made by Tesla is \frac{4}{13}.

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 66%.

Answer: 35

Let’s attempt this problem step-by-step. We’ll first determine the number of rows in tesla.merge(bmw, on="BodyStyle"), and then determine the number of rows in combo. For the purposes of the solution, let’s use temp to refer to the first merged DataFrame, tesla.merge(bmw, on="BodyStyle").

Recall, when we merge two DataFrames, the resulting DataFrame contains a single row for every match between the two columns, and rows in either DataFrame without a match disappear. In this problem, the column that we’re looking for matches in is "BodyStyle".

To determine the number of rows of temp, we need to determine which rows of tesla have a "BodyStyle" that matches a row in bmw. From the DataFrame provided, we can see that the only "BodyStyle"s in both tesla and bmw are SUV and sedan. When we merge tesla and bmw on "BodyStyle":

• The 4 SUV rows in tesla each match the 1 SUV row in bmw. This will create 4 SUV rows in temp.
• The 3 sedan rows in tesla each match the 1 sedan row in bmw. This will create 3 sedan rows in temp.

So, temp is a DataFrame with a total of 7 rows, with 4 rows for SUVs and 3 rows for sedans (in the "BodyStyle") column. Now, when we merge temp and audi on "BodyStyle":

• The 4 SUV rows in temp each match the 8 SUV rows in audi. This will create 4 \cdot 8 = 32 SUV rows in combo.
• The 3 sedan rows in temp each match the 1 sedan row in audi. This will create 3 \cdot 1 = 3 sedan rows in combo.

Thus, the total number of rows in combo is 32 + 3 = 35.

Note: You may notice that 35 is the result of multiplying the "SUV" and "Sedan" columns in the DataFrame provided, and adding up the results. This problem is similar to Problem 5 from the Fall 2021 Midterm.

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Difficulty: ⭐️⭐️⭐️⭐️

The average score on this problem was 45%.

Answer:

• Blank (a): np.arange(len(daily_conditions))
• Blank (b): days = days + 1
• Blank (c): [True, False]

At a high-level, here’s how this code block works:

• daily_conditions is an array of length 500, in each each element is True with probability 0.98 and False with probability 0.02. Each element of daily_conditions is meant to represent whether or not the given TritonTruck’s battery failed on that day. For instance, if the first four elements of daily_conditions are [True, True, False, True, ...], it means the battery was fine the first two days, but failed on the third day.
• The function days_until_replacement takes in daily_conditions and returns the number of days until the battery failed for the first time. In other words, it returns the number of elements before the first False in daily_conditions. In the example above, where the first four elements of daily_conditions are [True, True, False, True, ...], days_until_replacement would return 2, since the battery lasted 2 days until it needed to be replaced. It doesn’t matter what is in daily_conditions after the first False.

With that in mind, let’s fill in the pieces.

• Blank (a): We need to loop over all elements in daily_conditions. There are two ways to do this, in theory – by looping over the elements themselves (e.g. for cond in daily_conditions) or their positions (e.g. for i in np.arange(len(daily_conditions))). However, here we must loop over their positions, because the body of days_until_replacement uses daily_conditions[i], which only makes sense if i is the position of an element in daily_conditions. Since daily_conditions has 500 elements, the possible positions are 0, 1, 2, …, 499. Since len(daily_conditions) is 500, both np.arange(len(daily_conditions)) and np.arange(500) yield the same correct result here.

• Blank (b): days is the “counter” variable that is being used to keep track of the number of days the battery lasted before failing. If daily_conditions[i] is True, it means that the battery lasted another day without failing, and so 1 needs to be added to days. As such, the correct answer here is days = days + 1 (or days += 1). (If daily_conditions[i] is False, then the battery has failed, and so we return the number of days until the first failure.)

• Blank (c): This must be [True, False], as mentioned above. There are other valid answers too, including np.array([True, False]) and [1, 0].

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 56%.

Answer: 3

Here are two ways to solve this problem. In both solutions, we only look at Histogram A, since only Histogram A contains a bin that corresponds to EVs with exactly 6 "Seats". Recall, in histograms, bins are inclusive of the left endpoint and exclusive of the right endpoint, which means the [6, 7) bin represents EVs with >= 6 "Seats" and < 7 "Seats"; since the number of "Seats" is a whole number, this corresponds to exactly 6 "Seats".

Solution 1

Since the bin [6, 7) has a width of 1, its height is equal to its area, which is equal to the proportion of values in that bin. There are 32 values total, so all proportions (and, thus, the height of the [6, 7) bar) must be a multiple of \frac{1}{32}. The height is close to but just under 0.01; this implies the height is \frac{3}{32}, since that is also close to but just under 0.01 (\frac{4}{32} is way above and \frac{2}{32} is way below). So, we conclude that the number of EVs with 6 seats is 3.

Solution 2

The height of the [6, 7) bar is ever-so-slightly less than 0.1. If it’s height was 0.1, it would imply that the proportion of values in the [6, 7) bin was 0.1 \cdot (7 - 6) = 0.1, which would imply that the number of values in the [6, 7) bin is 0.1 \cdot 32 = 3.2. However, since the number of values in a bin must be an integer, the number of values in this bin is 3 (which is slightly less than 3.2).

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 61%.

Answer: 22

Now, we must look at Histogram B. In the previous part, we computed that there are 3 EVs with exactly 6 "Seats" in evs. Histogram B shows us the proportion, and thus number, of EVs with 5 or 6 "Seats", through its [5, 7) bin (remember, this bin corresponds to EVs with >= 5 "Seats" and < 7 "Seats"). If we can find the number of EVs with 5 or 6 "Seats", we can subtract 3 from it to determine the number of EVs with exactly 5 "Seats".

Since it’s not quite clear what the height of the [5, 7) bar is, we can create a range for the height of the bar, and use that to come up with a range for the area of the bar, and hence a range for the number of values in the bin. We can then use the fact that the number of values in a bin must be an integer to narrow down our answer.

• The height of the [5, 7) bar is less than 0.4. This means that the area of the [5, 7) bar is less than (7 - 5) \cdot 0.4 = 0.8, which means that the proportion of EVs with 5 or 6 "Seats" is less than 0.8, and the number of EVs with 5 or 6 "Seats" is less than 0.8 \cdot 32 = 25.6.
• The height of the [5, 7) bar is a bit above 0.375, which is the midpoint between 0.35 and 0.4 on the y-axis. This means that the area of the [5, 7) bar is more than (7 - 5) \cdot 0.375 = 0.75, which means that the number of EVs with 5 or 6 "Seats" is more than 0.75 \cdot 32 = 24.

We’ve found that the number of EVs with 5 or 6 "Seats" is more than 24 and less than 25.6. There is only one integer in this range – 25 – so the number of EVs with 5 or 6 "Seats" is 25. Finally, the number of EVs with exactly 5 seats is 25 - 3 = 22.

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Difficulty: ⭐️⭐️⭐️⭐️

The average score on this problem was 35%.

Answer: \frac{1}{5} (or 0.2)

Recall, the total area of a (density) histogram is 1. Since Histogram C only has one bar, the area of that one bar must be 1. we can use this fact to find what its height must be.

\begin{align*} \text{Area} &= \text{Width} \cdot \text{Height} \\ 1 &= (9 - 4) \cdot \text{Height} \\ \frac{1}{5} &= \text{Height} \end{align*}

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Difficulty: ⭐️⭐️⭐️

The average score on this problem was 68%.