Suppose that the distribution of math SAT scores from your state this year is normally distributed with mean 480 and standard deviation 100 for males, and mean 440 and standard deviation 120 for females. If someone who scores 780 or higher on the math SAT can be considered a genius, what is the proportion of geniuses among the male SAT takers?
A. 28%
B. 14%
C. 3%
D. 0.13%
E. 1.4%

The proportion of male geniuses among SAT takers, given a mean of 480 and a standard deviation of 100, with a genius score defined as 780 or higher, is approximately 0.13%.

To determine the proportion of male geniuses among the SAT takers, we first need to calculate a z-score for the given SAT score of 780. The z-score is a measure of how many standard deviations an element is from the mean. With a mean (μ) of 480 and a standard deviation (σ) of 100 for males, the z-score for a score of 780 is calculated as:

z = (X - μ) / σ = (780 - 480) / 100 = 3

Using the standard normal distribution, a z-score of 3 corresponds to a percentile that represents the area under the curve to the left of the z-score. This area can be found in standard normal distribution tables or using a calculator with statistical functions. The proportion of the area to the right of the z-score (which represents scores above 780) is what we are interested in. To find this, we subtract the area to the left from 1. The area to the left for a z-score of 3 is typically around 0.9987, thus the area to the right is: 1 - 0.9987 = 0.0013 or 0.13%

This suggests that approximately 0.13% of male SAT takers can be considered geniuses based on this criterion.

The proportion of geniuses among the male SAT takers is about 0.13%, which is found by calculating the z-score of 3 for a score of 780 and find that the proportion who score higher is 0.13%. The correct option is, therefore;

D. 0.13%

The steps used to find the proportion of geniuses among the male SAT takers can be presented as follows;

To calculate the proportion of geniuses among the male SAT takers, we need to determine the z-score for a score of 780 when the mean (μ) is 480 and the standard deviation (σ) is 100. The z-score is a measure of how many standard deviations an element is from the mean. The formula for the z-score is:

z = (X - μ) / σ

Where X is the score in question (780 in this case), μ is the mean, and σ is the standard deviation. Substituting the given values:

z = (780 - 480) / 100 = 3

A z-score of 3 indicates that a score of 780 is 3 standard deviations above the mean. Next, we use the standard normal distribution to find the proportion of scores above a z-score of 3. This can be found by looking at a z-table or using a statistical software. The area to the right of z = 3 (which corresponds to higher scores than 780) is approximately 1 - 0.99865 ≈ 0.00135, which is closest to the option 0.13%. Therefore, the proportion of male geniuses on the math SAT is 0.13%.

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