The admissions department is not only interested in point predictions for the verbal and mathematical intelligence scores but also requires an interval estimate for each score. This additional requirement ensures that the predicted range reflects the uncertainty in the model's predictions.

Once the University obtains the actual scores, participants will be evaluated based on two criteria:  
1. **Accuracy** — How close the predicted point estimates are to the true scores.  
2. **Coverage** — Whether the true score falls within the predicted interval.

To assess both aspects, the competition will use the **Mean Winkler Interval Score**. This scoring metric is defined as follows:

Given a prediction interval \\( [L\_i, U\_i] \\) for the true score \\( y\_i \\), with a target coverage probability of 95% (i.e., \\( 1 - \alpha = 0.95 \\)), the Winkler score for that observation is:

$$
W\_i = (U\_i - L\_i) + \frac{2}{\alpha} \max(0, L\_i - y\_i) + \frac{2}{\alpha} \max(0, y\_i - U\_i)
$$

The Mean Winkler Interval Score is the average of these scores across all observations:

$$
\text{Mean Winkler Interval Score} = \frac{1}{n} \sum\_{i=1}^n W\_i
$$

A lower score indicates better performance, rewarding both narrow intervals and correct coverage of the true values.

### Standardization of Scores
The predicted verbal and mathematical intelligence scores must be standardized on a scale with:  
- Mean = 100  
- Standard deviation = 10