The distribution of SAT scores of all college-bound seniors taking the SAT in 2014 was approximately normal with a mean [tex]\mu = 1497[/tex] and standard deviation [tex]\sigma = 322[/tex]. A certain test-retake preparation course is designed for students whose SAT scores are in the lower 25% of those who take the test in a given year. What is the maximum SAT score in 2014 that meets the course requirements?

To find the maximum SAT score for the lower 25% of students in 2014, the 25th percentile z-score of -0.675 is used, yielding a score of approximately 1279 when applying the normal distribution with a mean of 1497 and a standard deviation of 322.

Explanation:

The question pertains to finding the maximum SAT score in 2014 for students in the lower 25% percentile given a normal distribution of scores with a mean (μ) of 1497 and a standard deviation (σ) of 322.

To determine this, we need to find the z-score that corresponds to the 25th percentile in a normal distribution. This is typically found in z-score tables or by using a statistical calculator or software.

For the standard normal distribution, the z-score approximately corresponding to the 25th percentile is -0.675.

Once we have the z-score, we can use the formula:

SAT score = μ + (z × σ)

Substituting the given values:

SAT score = 1497 + (-0.675 × 322) ≈ 1497 - 217.85 ≈ 1279.15

Therefore, the maximum SAT score in 2014 that meets the course requirements for the lower 25% is approximately 1279.