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Answer :
Final answer:
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).
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