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To solve the problem of finding the proportion of students who scored over 514 on the MCAT, we will use the Z-score formula and standard normal distribution.

### Given:
- Mean score (\(\mu\)) = 500.9
- Standard deviation (\(\sigma\)) = 10.6
- Score of interest = 514

### Step 1: Calculate the Z-score
The Z-score formula is:
\[
Z = \frac{X - \mu}{\sigma}
\]
where:
- \(X\) is the score of interest (514),
- \(\mu\) is the mean (500.9),
- \(\sigma\) is the standard deviation (10.6).

Substituting the given values:

\[
Z = \frac{514 - 500.9}{10.6}
\]

Let me calculate this value.The Z-score is approximately 1.24.

### Step 2: Find the proportion from the Z-table
Using a standard normal distribution table (Z-table), we will look up the proportion corresponding to a Z-score of 1.24.

The Z-table gives the area to the left of a given Z-score. For \( Z = 1.24 \), the table shows a cumulative probability of approximately 0.8925.

### Step 3: Calculate the proportion of students with scores over 514
To find the proportion of students who scored over 514, we subtract this cumulative probability from 1:
\[
P(X > 514) = 1 - 0.8925 = 0.1075
\]

### Step 4: Convert to percentage
\[
\text{Percentage} = 0.1075 \times 100 = 10.75\%
\]

So, approximately **10.75%** of students scored over 514.

Would you like more details on any part of this solution? Here are some related questions for deeper exploration:

1. How is the Z-score used in normal distribution problems?
2. What does the cumulative probability in a Z-table represent?
3. How do you interpret the area under the normal curve?
4. How would this calculation change if the standard deviation were different?
5. How can the Z-score method be used for other types of standardized tests?

**Tip:** When using Z-tables, always remember that they give the area to the left of the Z-score, so for questions asking for values above a certain point, subtract the cumulative probability from 1.

What proportion of students taking the MCAT had a score over 514? Compute the proportion and then enter the answer as a percentage rounded to two decimal places.

Learn how to calculate the proportion of students who scored over 514 on the MCAT using Z-scores and standard normal distribution. This detailed guide walks you through each step, including calculating the Z-score and using the Z-table for final results. Ideal for undergraduate statistics students.

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