Question

Your company administers an executive aptitude test. They report test grades as z-scores, and you got a score of 2.30. They will admit to the executive training program only people who score in the top 3% on this test.

a) With your z-score of 2.30, did you make the cut?
b) What do you need to assume about test scores to find your answer in part a?

Answer 1

With a z-score of 2.30, you have made the cut for the top 3% for the executive training program, since you need a z-score higher than about 1.88 to be in the top 3%. Z-scores are valid under the assumption of normal distribution of the test scores.

Step-by-step explanation:

With a z-score of 2.30, you are indeed within the top 3% of the test scores. To determine this, we look at the empirical rule, also known as the 68-95-99.7 rule, which states that approximately 95% of data falls within two standard deviations (-2 and +2 z-scores) and 99.7% within three standard deviations (-3 and +3 z-scores) from the mean. Since a z-score of 2 is approximately at the 97.5th percentile and a score of 3 is well above the 99.7th percentile, a z-score of 2.30 means you are between these percentiles. Therefore, being in the top 3% suggests that you need a z-score greater than approximately 1.88, so a z-score of 2.30 is more than sufficient to qualify for the executive training program.

For part b, we need to assume that the test scores are normally distributed to apply z-scores accurately. This is a basic requirement for the empirical rule to be valid and to interpret z-scores in the context of percentages and percentiles.