Question

if your score on your next statistics test is converted to a z score, which of these z scores would you prefer: - 2.00, - 1.00, 0, 1.00, 2.00? why?
a. the z score of - 1.00 is most preferable because it is 1.00 standard deviation below the mean and would correspond to an above average test score.
b. the z score of 0 is most preferable because it corresponds to a test score equal to the mean.
c. the z score of - 2.00 is most preferable because it is 2.00 standard deviations below the mean and would correspond to the highest of the five different possible test scores.
d. the z score of 2.00 is most preferable because it is 2.00 standard deviations above the mean and would correspond to the highest of the five different possible test scores.
e. the z score of 1.00 is most preferable because it is 1.00 standard deviation above the mean and would correspond to an above average test score.

Explanation:

Step1: Understand z - score concept

A z - score of \(z=\frac{x - \mu}{\sigma}\), where \(x\) is the raw score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. A positive z - score means the score is above the mean, and a negative z - score means the score is below the mean. The magnitude of the z - score represents the number of standard deviations from the mean.

Step2: Analyze each option

• Option A (\(z=- 1.00\)): A z - score of \(-1.00\) means the score is \(1\) standard deviation below the mean.
• Option B (\(z = 0\)): A z - score of \(0\) means the score is equal to the mean.
• Option C (\(z=-2.00\)): A z - score of \(-2.00\) means the score is \(2\) standard deviations below the mean.
• Option D (\(z = 2.00\)): A z - score of \(2.00\) means the score is \(2\) standard deviations above the mean. Since \(2>1>0\) (in terms of magnitude for positive z - scores, as we want a high score), a \(z = 2.00\) indicates a higher score relative to the mean compared to \(z=1.00\) (Option E)

Answer:

D. The z score of 2.00 is most preferable because it is 2.00 standard deviations above the mean and would correspond to the highest of the five different possible test scores.