Final answer:
To be eligible for a $500 award in a test with a mean score of 400 and a standard deviation of 100, a student would need to score at least 700, which corresponds to the z-score for the top 0.15% of students.
Step-by-step explanation:
To find the minimum score you would need to receive a $500 award on a high school mandatory test with a mean of 400 and a standard deviation of 100, we need to determine the z-score that corresponds to the top 0.15% of students. This is a problem involving normal distribution and z-scores.
First, we look up the z-score that corresponds to the top 0.15% in a standard normal distribution table, or we can use a calculator or software that provides this functionality.
Typically, for the top 0.15%, the z-score is about 3.0 (this value might slightly vary depending on the source).
Now, we can use the z-score formula:
Z = (X - μ) / σ
Where Z is the z-score, X is the test score, μ is the mean score, and σ is the standard deviation.
Solving for X gives us:
X = Z * σ + μ
Substituting the known values:
X = 3.0 * 100 + 400
= 700
Therefore, a student would need a score of at least 700 to be in the top 0.15% of the test takers and to receive the $500 award.