Question

if a data value in a normal distribution has a negative z-score, which of the following must be true?
the data value must be negative.
the data value must be positive.
the data value must be less than the mean.
the data value must be greater than the mean.

Explanation:

The z-score formula is $z = \frac{x - \mu}{\sigma}$, where $x$ is the data value, $\mu$ is the population mean, and $\sigma$ (the standard deviation) is always positive. A negative z-score means $z < 0$. Substituting the formula, $\frac{x - \mu}{\sigma} < 0$. Since $\sigma > 0$, multiplying both sides by $\sigma$ preserves the inequality, giving $x - \mu < 0$, so $x < \mu$. The sign of $x$ itself does not determine the z-score, only its relation to the mean.

Answer:

The data value must be less than the mean.