Question

An aptitude test is designed to measure leadership abilities of the test subjects. Suppose that the scores on the test are normally distributed with a mean of 550 and a standard deviation of 125. The individuals who exceed 780 on this test are considered to be potential leaders. What proportion of the population are considered to be potential leaders? Round your answer to at least four decimal places.

Answer 1

Using the normal distribution, it is found that 0.0329 = 3.29% of the population are considered to be potential leaders.

In a normal distribution with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

• It measures how many standard deviations the measure is from the mean.  

• After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.

In this problem:

• The mean is of 550, hence $\mu = 550$.

• The standard deviation is of 125, hence $\sigma = 125$.

The proportion of the population considered to be potential leaders is 1 subtracted by the p-value of Z when X = 780, hence:

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{780 - 550}{125}$

$Z = 1.84$

$Z = 1.84$ has a p-value of 0.9671.

1 - 0.9671 = 0.0329

0.0329 = 3.29% of the population are considered to be potential leaders.

To learn more about the normal distribution, you can take a look at brainly.com/question/24663213