Final answer:
The probability of scoring over 650 is 6.68%, the probability of scoring less than 459 is 33.94%, the probability of scoring between 325 and 675 is 81.86%, the proportion of the student's pool that would be eligible for admission with a score over 680 is 3.59%, and the score that makes the top 20% of the students eligible is 584.
Step-by-step explanation:
a) To find the probability that a student will score over 650, we need to calculate the z-score and use the z-table. The z-score is calculated as (x - mean) / standard deviation. So, the z-score for a score of 650 is (650 - 500) / 100 = 1.5. Using the z-table, we find that the probability of getting a score above 1.5 standard deviations is approximately 0.0668 or 6.68%.
b) To find the probability that a student will score less than 459, we calculate the z-score as (459 - 500) / 100 = -0.41. Using the z-table, we find that the probability of getting a score below -0.41 standard deviations is approximately 0.3394 or 33.94%.
c) To find the probability that a student will score between 325 and 675, we calculate the z-scores for 325 and 675. The z-score for 325 is (325 - 500) / 100 = -1.75, and the z-score for 675 is (675 - 500) / 100 = 1.75. Using the z-table, we find that the probability of getting a score between -1.75 and 1.75 standard deviations is approximately 0.8186 or 81.86%.
d) If a school only admits students who score over 680, we calculate the z-score for 680 as (680 - 500) / 100 = 1.8. Using the z-table, we find that the proportion of the student's pool that would be eligible for admission is approximately 0.0359 or 3.59%.
e) To find the score that makes the top 20% of the students eligible, we need to find the z-score corresponding to the 80th percentile. Using the z-table, we find that the z-score for the 80th percentile is approximately 0.84. We can calculate the score using the formula score = (z-score * standard deviation) + mean. So, the score that makes the top 20% of the students eligible is (0.84 * 100) + 500 = 584.