Final answer:
This high school level Mathematics question involves using statistics and probability to determine the probability of a single SAT score and a sample mean of SAT scores exceeding 1600, applying the Central Limit Theorem for the latter scenario.
Step-by-step explanation:
The subject of this two-part question is Mathematics, specifically statistics and probability as it pertains to the Central Limit Theorem (CLT). To solve the problem, we'll address the two separate scenarios.
1. Probability of a Single Score Exceeding 1600
First, we find the probability of a score exceeding 1600 when scores are normally distributed with a mean (μ) of 1518 and a standard deviation (σ) of 325. We compute the z-score for 1600 and use the standard normal distribution to find the probability.
2. Probability of Sample Mean Exceeding 1600
Second, we apply the Central Limit Theorem for a sample of 81 SAT scores to find the probability that their mean score exceeds 1600. The key is that as the sample size increases, the distribution of sample means approaches a normal distribution even if the population distribution is not normal.
The Central Limit Theorem ensures the distribution of the sample means will be normal with a mean equal to the population mean and a standard deviation equal to σ/√n (σ is the population standard deviation and n is the sample size). We find the z-score for a sample mean of 1600 and then calculate the corresponding probability.