Question
he 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 find the expected score of a student on the AP Literature Exam, we need to calculate the weighted average of the scores using the distribution percentages.
### Step 1: Convert Percentages to Proportions
We first convert the percentages into proportions (decimals) for easier calculations:
- Score 1: 10.4% = 0.104
- Score 2: 31.7% = 0.317
- Score 3: 31.5% = 0.315
- Score 4: 18.8% = 0.188
- Score 5: 7.6% = 0.076
### Step 2: Calculate the Expected Score
The expected score \( E(X) \) can be calculated using the formula:
\[
E(X) = \sum (x_i \cdot p_i)
\]
Where \( x_i \) is the score and \( p_i \) is the proportion of students receiving that score.
Now we substitute in the values:
\[
E(X) = (1 \cdot 0.104) + (2 \cdot 0.317) + (3 \cdot 0.315) + (4 \cdot 0.188) + (5 \cdot 0.076)
\]
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:
\[
E(X) = 0.104 + 0.634 + 0.945 + 0.752 + 0.380
\]
Calculating the total:
\[
E(X) = 0.104 + 0.634 = 0.738
\]
\[
E(X) = 0.738 + 0.945 = 1.683
\]
\[
E(X) = 1.683 + 0.752 = 2.435
\]
\[
E(X) = 2.435 + 0.380 = 2.815
\]
Thus, the expected score is
\[
E(X) = 2.815.
\]
### Interpretation in Context
The expected score of approximately 2.82 (which we can round to 2.8 for practical purposes) indicates that, on average, a student taking the AP Literature Exam is projected to score slightly below a 3 (the threshold for "qualified" status on many AP exams). This means that while a significant number of students achieve a score of 3 or higher, the average score reflects a slightly lower performance overall, suggesting that many students may be struggling to reach the passing threshold of 3 (which is generally considered sufficient for college credit).
### Step 1: Convert Percentages to Proportions
We first convert the percentages into proportions (decimals) for easier calculations:
- Score 1: 10.4% = 0.104
- Score 2: 31.7% = 0.317
- Score 3: 31.5% = 0.315
- Score 4: 18.8% = 0.188
- Score 5: 7.6% = 0.076
### Step 2: Calculate the Expected Score
The expected score \( E(X) \) can be calculated using the formula:
\[
E(X) = \sum (x_i \cdot p_i)
\]
Where \( x_i \) is the score and \( p_i \) is the proportion of students receiving that score.
Now we substitute in the values:
\[
E(X) = (1 \cdot 0.104) + (2 \cdot 0.317) + (3 \cdot 0.315) + (4 \cdot 0.188) + (5 \cdot 0.076)
\]
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:
\[
E(X) = 0.104 + 0.634 + 0.945 + 0.752 + 0.380
\]
Calculating the total:
\[
E(X) = 0.104 + 0.634 = 0.738
\]
\[
E(X) = 0.738 + 0.945 = 1.683
\]
\[
E(X) = 1.683 + 0.752 = 2.435
\]
\[
E(X) = 2.435 + 0.380 = 2.815
\]
Thus, the expected score is
\[
E(X) = 2.815.
\]
### Interpretation in Context
The expected score of approximately 2.82 (which we can round to 2.8 for practical purposes) indicates that, on average, a student taking the AP Literature Exam is projected to score slightly below a 3 (the threshold for "qualified" status on many AP exams). This means that while a significant number of students achieve a score of 3 or higher, the average score reflects a slightly lower performance overall, suggesting that many students may be struggling to reach the passing threshold of 3 (which is generally considered sufficient for college credit).