Question
The table below shows the distribution of scores on the AP Literature Exam. What is a student's expected score? What does this value mean in context? Show all of your work.
Score 1 2 3 4 5
% Receiving 10.4 31.7 31.5 18.8 7.6
Score 1 2 3 4 5
% Receiving 10.4 31.7 31.5 18.8 7.6
Answers
GPT-4o mini
To calculate a student's expected score for the AP Literature Exam, we will use the formula for the expected value \( E(X) \):
\[
E(X) = \sum (x_i \cdot p_i)
\]
Where:
- \( x_i \) is the score,
- \( p_i \) is the probability of receiving that score (which we derive from the percentages).
First, we convert the percentages into probabilities:
- For a score of 1: \( p(1) = 0.104 \)
- For a score of 2: \( p(2) = 0.317 \)
- For a score of 3: \( p(3) = 0.315 \)
- For a score of 4: \( p(4) = 0.188 \)
- For a score of 5: \( p(5) = 0.076 \)
Now we will multiply each score by its corresponding probability:
\[
E(X) = (1 \cdot 0.104) + (2 \cdot 0.317) + (3 \cdot 0.315) + (4 \cdot 0.188) + (5 \cdot 0.076)
\]
Calculating each term:
- \( 1 \cdot 0.104 = 0.104 \)
- \( 2 \cdot 0.317 = 0.634 \)
- \( 3 \cdot 0.315 = 0.945 \)
- \( 4 \cdot 0.188 = 0.752 \)
- \( 5 \cdot 0.076 = 0.380 \)
Now we sum these values:
\[
E(X) = 0.104 + 0.634 + 0.945 + 0.752 + 0.380
\]
\[
E(X) = 2.815
\]
Thus, the expected score \( E(X) \) is approximately 2.82.
### Interpretation:
The expected score of approximately 2.82 means that, on average, a student who takes the AP Literature Exam can expect to score just below a 3. This value is significant because it gives insight into the overall performance of students on the exam. A score of 3 is typically considered passing, suggesting that most students in this dataset are likely to score around or below the average pass mark, with many falling into the lower score categories. The distribution indicates varied levels of performance among test-takers.
\[
E(X) = \sum (x_i \cdot p_i)
\]
Where:
- \( x_i \) is the score,
- \( p_i \) is the probability of receiving that score (which we derive from the percentages).
First, we convert the percentages into probabilities:
- For a score of 1: \( p(1) = 0.104 \)
- For a score of 2: \( p(2) = 0.317 \)
- For a score of 3: \( p(3) = 0.315 \)
- For a score of 4: \( p(4) = 0.188 \)
- For a score of 5: \( p(5) = 0.076 \)
Now we will multiply each score by its corresponding probability:
\[
E(X) = (1 \cdot 0.104) + (2 \cdot 0.317) + (3 \cdot 0.315) + (4 \cdot 0.188) + (5 \cdot 0.076)
\]
Calculating each term:
- \( 1 \cdot 0.104 = 0.104 \)
- \( 2 \cdot 0.317 = 0.634 \)
- \( 3 \cdot 0.315 = 0.945 \)
- \( 4 \cdot 0.188 = 0.752 \)
- \( 5 \cdot 0.076 = 0.380 \)
Now we sum these values:
\[
E(X) = 0.104 + 0.634 + 0.945 + 0.752 + 0.380
\]
\[
E(X) = 2.815
\]
Thus, the expected score \( E(X) \) is approximately 2.82.
### Interpretation:
The expected score of approximately 2.82 means that, on average, a student who takes the AP Literature Exam can expect to score just below a 3. This value is significant because it gives insight into the overall performance of students on the exam. A score of 3 is typically considered passing, suggesting that most students in this dataset are likely to score around or below the average pass mark, with many falling into the lower score categories. The distribution indicates varied levels of performance among test-takers.