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Answer :
The TOEFL score that separates the top 25% and bottom 75% is approximately 96.1. And the probability that an international candidate who scores above average will also meet the minimum TOEFL score requirement of 108 is 0.048 or 4.78%.
Firstly, let's look at the score that separates the top 25% and the bottom 75%. This is often referred to as the 75th percentile. Since the scores follow a normal distribution, we use a standard z-score table to find the corresponding z-score for the top 25%, which is approximately 0.675. We then convert this z-score to a TOEFL score using the formula: TOEFL Score = Mean + (Z-score * Standard Deviation). Given the mean is 88 and the standard deviation is 12, our calculation becomes: TOEFL Score = 88 + (0.675 * 12), which equals around 96.1. This means option A. 96.0939 is the closest answer.
Next, let's address the second part of your question. Here, we need to calculate the probability that a candidate who scores above average will also meet the minimum TOEFL score requirement of 108 set by the education department. To do this, we need to calculate a new z-score: Z = (108 - Mean) / Standard Deviation. Plugging in our values, we get Z = (108 - 88) / 12 = 1.67. Looking up 1.67 in the z-score table, we see the area to the left is 0.9522, which means there is a 95.22% chance that a randomly chosen candidate would score less than 108. To find the probability that the candidate scores 108 or more, we subtract this from 1 (because the total probability is 1), which gives us 1 - 0.9522 = 0.0478 or 4.78%. So the answer is B. 0.048.
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