Answer: 68%
Explanation:
To find the percentage of students who scored between 588 and 680 on the exam, we can use the properties of the normal distribution and z-scores.
1. **Calculate Z-scores:**
The formula for calculating the z-score is: \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
For 588:
\(z_{588} = \frac{588 - 588}{46} = 0\)
For 680:
\(z_{680} = \frac{680 - 588}{46} \approx 2.00\)
2. **Find the Percentage:**
To find the percentage between these two z-scores, we can use a standard normal distribution table (also known as a Z-table).
The Z-table provides the cumulative probability up to a certain z-score. Looking up the z-score of 2.00 in the Z-table, we find that the cumulative probability is approximately 0.9772.
The cumulative probability up to the z-score of 0 (which corresponds to the mean) is 0.5000.
3. **Calculate the Percentage:**
The percentage of students scoring between 588 and 680 is the difference between these two cumulative probabilities:
Percentage = \(0.9772 - 0.5000 \approx 0.4772\)
To express this as a percentage, we multiply by 100:
Percentage ≈ \(0.4772 \times 100 \approx 47.72\)%.
Therefore, approximately 47.72% of the students scored between 588 and 680 on the exam.