High school students across the nation compete in a financial capability challenge each year by taking a National Financial Capability Challenge Exam. Students who score in the top 25 percent are recognized publicly for their achievement by the Department of the Treasury. Assuming a normal distribution, how many standard deviations above the mean does a student have to score to be publicly recognized?

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joaobezerra joaobezerra · Answer 1 · 2021-12-07T11:19:43+01:00

Using the normal distribution, it is found that a student has to score 0.675 standard deviations above the mean to be publicly recognized.

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

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

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.

The top 25% is at least the 100 - 25 = 75th percentile, which is X when z has a p-value of 0.75.

Hence, a student has to score 0.675 standard deviations above the mean to be publicly recognized.

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