7. For a given population of high school seniors, the SAT scores in mathematics are normally distributed with a mean score of 500 and a standard deviation of 100. Suppose Berkeley College will accept only high school seniors with a mathematics SAT score in the top $5 \%$. What is the cutoff SAT score needed in mathematics to be accepted into Berkeley?

Step 1 ：The problem is asking for the cutoff SAT score needed in mathematics to be accepted into Berkeley College, which only accepts the top 5% of high school seniors based on their mathematics SAT scores. The scores are normally distributed with a mean of 500 and a standard deviation of 100.

Step 2 ：To solve this problem, we need to find the z-score that corresponds to the top 5% of the distribution. The z-score formula is: \(Z = \frac{X - \mu}{\sigma}\), where \(Z\) is the z-score, \(X\) is the value from the data, \(\mu\) is the mean and \(\sigma\) is the standard deviation.

Step 3 ：In this case, we know that \(\mu = 500\) and \(\sigma = 100\). We need to find the z-score that corresponds to the top 5% of the distribution and then solve for \(X\).

Step 4 ：The z-score that corresponds to the top 5% of the distribution is approximately 1.645.

Step 5 ：Substituting the values into the z-score formula, we get \(1.645 = \frac{X - 500}{100}\). Solving for \(X\), we find that \(X = 1.645 \times 100 + 500\), which simplifies to \(X \approx 664.49\).

Step 6 ：Therefore, the cutoff SAT score needed in mathematics to be accepted into Berkeley College is approximately \(\boxed{664.49}\). This means that a student needs to score at least this much to be in the top 5% of the distribution and be accepted into Berkeley.