If a z-score is zero, which of the following must be true? Explain your reasoning.
- The mean is zero.
- The corresponding $x$-value is zero
- The corresponding $x$-value is equal to the mean.
Choose the correct answer below.
A. The mean is zero, because the mean is always equal to the z-score.
B. The corresponding $x$-value is zero, because the $z$-score is equal to the $x$-value divided by the standard deviation.
C. The corresponding $x$-value is equal to the mean, because the $z$-score is equal to the difference between the $x$-value and the mean, divided by the standard deviation.
Answer
\(\boxed{\text{C. The corresponding $x$-value is equal to the mean, because the $z$-score is equal to the difference between the $x$-value and the mean, divided by the standard deviation.}}\)
Step 1 ：The z-score is a measure of how many standard deviations an element is from the mean.
Step 2 ：If the z-score is zero, it means that the element is exactly at the mean.
Step 3 ：Therefore, the corresponding x-value is equal to the mean.
Step 4 ：The mean itself is not necessarily zero, and the x-value is not necessarily zero either. They are just equal to each other.
Step 5 ：\(\boxed{\text{C. The corresponding $x$-value is equal to the mean, because the $z$-score is equal to the difference between the $x$-value and the mean, divided by the standard deviation.}}\)
