The SAT math scores for applicants to a particular engineering school are normally distributed with a mean of 680 and a standard deviation of 35. Suppose that only applicants with scores above 700 are considered for admission. What percentage of the applicants considered have scores below 750?

1) 2.3
2) 28.4
3) 71.6
4) 92.0
5) 97.7

To find the percentage of SAT applicants with math scores below 750 from those already considered for having over 700, we must calculate z-scores and use the standard normal distribution. Approximately 92% of these applicants have scores below 750, corresponding to option 4 on the list provided.

Explanation:

The question asks about the percentage of applicants with SAT math scores below 750, given that they are already above 700, for a normally distributed set of scores with a mean of 680 and a standard deviation of 35. Since the distribution is normal, this can be found using z-scores and the standard normal distribution table.

First, we find the z-score for 700: z = (700 - 680) / 35 = 20/35 ≈ 0.57.

Next, we find the z-score for 750: z = (750 - 680) / 35 = 70/35 ≈ 2.00.

Looking up these z-scores in a standard normal distribution table or using a calculator gives us the cumulative probabilities. For z = 0.57, the cumulative probability is about 0.7157 (71.57%), and for z = 2.00, it is about 0.9772 (97.72%).

To find the percentage of applicants with scores between 700 and 750, we subtract the cumulative probability for z = 0.57 from the cumulative probability for z = 2.00.
Percentage = (0.9772 - 0.7157) × 100 ≈ 26.15%

However, since we are only considering applicants with scores above 700, we need to adjust this percentage to the subset above 700. To find this, we take 1 minus the probability of z = 0.57, which is 1 - 0.7157 = 0.2843 (28.43%). Then, we divide 26.15% by 28.43% to get the relative proportion within this subset.
Final Percentage = (26.15 / 28.43) × 100 ≈ 91.96%

The correct answer is therefore close to 92%, which corresponds to option 4).