Question

the grades of a science test are normally distributed. thanh finds that his grade on the test has a z-score of \\(-2.5\\).
which statement must be true?

thanhs grade is 2.5 standard deviations below the mean of the test grades.
thanhs grade is 2.5 points below the mean of the test grades.
thanhs grade is 2.5 standard deviations above the mean of the test grades.
thanhs grade is 2.5 points above the mean of the test grades.

Explanation:

Define the meaning of a z-score

Using the Z-Score Formula knowledge point

$$ z = \frac{x - \mu}{\sigma} $$

where \(x\) is the data value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. A z-score measures the number of standard deviations a data point is from the mean.

Interpret the given z-score value

Using the Z-Score Formula knowledge point

$$ z = -2.5 $$

The negative sign indicates that the value is below the mean, and the magnitude \(2.5\) indicates it is exactly \(2.5\) standard deviations away. Thus, Thanh's grade is \(2.5\) standard deviations below the mean.

Answer:

• Thanh's grade is 2.5 standard deviations below the mean of the test grades. (Correct answer)
• Thanh's grade is 2.5 points below the mean of the test grades.
• Thanh's grade is 2.5 standard deviations above the mean of the test grades.
• Thanh's grade is 2.5 points above the mean of the test grades.