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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.
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