Final answer:
A z-score of zero indicates that the corresponding x-value is equal to the mean of the distribution. It does not imply that the mean or the x-value itself is zero; rather, it shows that the x-value is at the mean of its distribution.
Step-by-step explanation:
If a z-score is zero, it must be true that the corresponding x-value is equal to the mean. This is because a z-score represents the number of standard deviations a data point is from the mean of a distribution. Given the standard normal distribution, represented by Z ~ N(0, 1), the mean is 0 and the standard deviation is 1. Therefore, if z=0, it indicates that x is exactly at the mean of the original distribution, not that the mean or the x-value itself is zero.
For example, if a certain random variable X follows a normal distribution N(µ, σ) and we have an x-value that is equal to µ, the z-score for this x-value would be zero. This is because the z-score formula (z = (x - µ) / σ) shows that when x is µ, the formula becomes (µ - µ) / σ = 0 / σ = 0. A z-score of zero always corresponds to an x-value that is exactly the mean of the distribution, regardless of what the actual mean (µ) or standard deviation (σ) are.