Answer :
Final answer:
In a normal distribution, the 25th percentile, or the first quartile, is typically represented by a z-score of -0.674. Using the given mean and standard deviation values of the SAT scores, a student that scores 1400 or less is in the bottom 25% and qualifies for the test retake preparation course.
Explanation:
The given question requires us to understand the normal distribution in statistics and how percentile scores are calculated. If this SAT test preparation course is designed for the lower 25% of test-takers, we need to identify the SAT score that would cut-off the bottom 25% of scores (the first quartile or the 25th percentile). In a normal distribution, this is often represented by a z-score, which measures how many standard deviations an element is from the mean.
In this scenario, we have a mean SAT score (μ) of 1497 and a standard deviation (σ) of 322. The z-score associated with the 25th percentile in a standard normal distribution is roughly -0.674. Therefore, the SAT score (X) that represents the 25th percentile can be calculated using the formula X = μ + zσ. When we substitute the known values into this formula, it implies that X = 1497 + (-0.674)*322, approximately equal to 1497 - 217 = 1280.
Thus, a student with an SAT score of 1280 or lower would typically be eligible for the test-retake preparation course. From the options provided, the only score less than 1280 is 1400. Hence the correct answer is option (A) 1400.
Learn more about SAT Scores and Normal Distribution here:
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