Final answer:
To find the minimum SAT score to be in the top 25%, we determine the z-score for the 75th percentile, approximately 0.674, and use the formula X = Zσ + μ to calculate the score, which is around 567.
Step-by-step explanation:
The student's question is related to the normal distribution and percentile ranking in the context of SAT scores. To find the minimum score necessary to be in the top 25% of the SAT distribution, we start by noting that the top 25% corresponds to the highest quartile, which is above the 75th percentile. In a normal distribution, percentiles correspond to specific z-scores, where a z-score is the number of standard deviations a value is from the mean.
Using a standard normal distribution table or a calculator, we find that the z-score for the 75th percentile is approximately 0.674. Hence, to calculate the minimum score necessary for the top 25%, we use the following formula:
- Z = (X - μ) / σ
- X = Zσ + μ
Replacing the z-score, mean (μ = 500), and standard deviation (σ = 100) values into the formula, we get:
X = 0.674(100) + 500 = 567.4
Therefore, the minimum SAT score necessary to be in the top 25% is approximately 567.