Final answer:
To find where 75% of the population scores above on an IQ test with average of 100 and standard deviation of 15, we look for the 25th percentile below the mean but less than one standard deviation away. The correct answer falls below the mean and is closest to one standard deviation below the mean, which is option a), 85.
Step-by-step explanation:
The question asks for the IQ score above which 75% of the population lies, given a normally distributed set of IQ scores with a mean of 100 and a standard deviation of 15.
To solve this, we need to find the value on the normal distribution curve for which the area to the right (representing the percentage of the population with higher IQ scores) is 0.75. This corresponds to the 25th percentile, also known as the first quartile, as the question is asking for the lower boundary above which the top 75% of scores lie.
Using the properties of the standard normal distribution, we know that the 50th percentile is the mean, which is an IQ score of 100. Since the distribution is symmetric, the 25th percentile will be below this value.
One standard deviation below the mean spans 68% of the center of the distribution, therefore the 25th percentile will be less than one standard deviation below the mean.
We are looking for a value that is less than the mean but not by a full standard deviation. Looking at the options given and knowing that one standard deviation below the mean would be 85 (100 - 15), the only possible value that makes sense and is less than the mean but closer to it than 85 is option a), 85.