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Answer :
Final answer:
The middle 68% of SAT scores ranged from 1317 to 1705, the middle 95% from 1123 to 1899, and to be in the top 2.5%, a student needs to score about 1899 or above.
Explanation:
To answer the student's questions about SAT scores:
(a) Range of the Middle 68% of SAT Scores
The middle 68% of scores in a normal distribution falls within one standard deviation of the mean. Hence, the range is:
Upper bound = Mean + Standard deviation = 1511 + 194 = 1705
Therefore, the middle 68% of SAT scores ranged from 1317 to 1705.
Thus, the range for the middle 95% of SAT scores was from 1123 to 1899.
(c) Score to be in the Top 2.5% of SAT Scores
To be in the top 2.5%, a student would need a Z score that corresponds to the 97.5th percentile. This is approximately a Z score of +2 standard deviations. Thus, the score would be:
Score = Mean + Z*Standard deviation = 1511 + 2*194 ≈ 1899
A student must score approximately 1899 or above to be in the top 2.5% of SAT scores.
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