if a data value in a normal distribution has a negative z-score, which of the following must be true?\nthe…

Explanation:

Step1: Recall Z - score formula

The formula for the z - score is (z=\frac{x - \mu}{\sigma}), where (x) is the data value, (\mu) is the mean of the distribution, and (\sigma) is the standard deviation (and (\sigma>0) since it is a measure of spread).

Step2: Analyze negative z - score

If (z<0), then from the formula (z = \frac{x-\mu}{\sigma}), since (\sigma>0), we can multiply both sides of the inequality (z<0) by (\sigma) (which doesn't change the direction of the inequality) to get (x - \mu<0). Then, by adding (\mu) to both sides of the inequality (x-\mu < 0), we obtain (x<\mu). This means that the data value (x) is less than the mean (\mu).

Now let's analyze the other options:

• Option "The data value must be negative": The z - score depends on the difference between the data value and the mean relative to the standard deviation. For example, if the mean (\mu = 10) and (\sigma=5), a data value (x = 8) has a z - score (z=\frac{8 - 10}{5}=\frac{- 2}{5}=- 0.4) (negative z - score) but the data value (x = 8) is positive. So this option is incorrect.
• Option "The data value must be positive": Using the same example as above, if (\mu = 3) and (\sigma = 1), a data value (x=1) has a z - score (z=\frac{1 - 3}{1}=-2) (negative z - score) and the data value (x = 1) is positive? No, (x = 1) is positive? Wait, no, (x = 1) is positive, but if (\mu=-1) and (\sigma = 1), a data value (x=-3) has a z - score (z=\frac{-3-(-1)}{1}=\frac{-2}{1}=-2) (negative z - score) and the data value is negative. So a data value with a negative z - score can be positive or negative, so this option is incorrect.
• Option "The data value must be greater than the mean": From our earlier derivation, we found that (x<\mu) when (z < 0), so this option is incorrect.

Answer:

The data value must be less than the mean. (The option corresponding to "The data value must be less than the mean")