In aptitude test is designed to measure leadership abilities of the test subjects suppose that the scores on the test are normally distributed with the mean of 570 and a standard deviation of 125. the individuals who exceed 750 on this test are considered to be potentially leaders. what proportion of the population are considered to be a potential leaders? round your answer to that is for decimal places 

To determine the proportion of the population considered to be potential leaders (those who score above 750 on the test), we need to calculate the area under the normal distribution curve to the right of \( X = 750 \). This involves finding the corresponding \( Z \)-score and then using the standard normal distribution.

### Step 1: Calculate the \( Z \)-score
The \( Z \)-score is calculated using the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

where:
- \( X = 750 \) (the score threshold for potential leaders)
- \( \mu = 570 \) (the mean score)
- \( \sigma = 125 \) (the standard deviation)

\[
Z = \frac{750 - 570}{125} = \frac{180}{125} = 1.44
\]

### Step 2: Find the area to the right of \( Z = 1.44 \)
We need to find the proportion of the population that has a \( Z \)-score greater than 1.44. This is done by finding the cumulative probability up to \( Z = 1.44 \) and then subtracting that value from 1.

Using the standard normal distribution table or a calculator:

- The cumulative probability up to \( Z = 1.44 \) is approximately 0.9251.

So, the area to the right of \( Z = 1.44 \) is:

\[
P(X > 750) = 1 - 0.9251 = 0.0749
\]

### Conclusion
The proportion of the population that is considered to be potential leaders is **0.0749** or **7.49%**.

Would you like more details or have any other questions?

Here are some related questions you might consider:
1. What is the \( Z \)-score for a score of 650 on this test?
2. How would the proportion of potential leaders change if the standard deviation were 100 instead of 125?
3. What is the probability that a randomly selected individual scores below 500 on this test?
4. How would you find the proportion of individuals scoring between 600 and 700?
5. What is the probability of scoring exactly 570 on this test?
6. How does the mean and standard deviation affect the shape of the normal distribution?
7. What score corresponds to the top 5% of the population?
8. How do you use the normal distribution to assess test results in other scenarios?

**Tip:** When working with normal distributions, always start by finding the \( Z \)-score, as it allows you to standardize the problem and use the standard normal distribution table.

Learn how to calculate the proportion of potential leaders in a normal distribution with a mean of 570 and a standard deviation of 125. This problem involves using the Z-score and cumulative probability to determine the proportion of individuals scoring above 750. Suitable for advanced high school students and beyond.

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