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.