To determine which data point results in a z-score that indicates it is 2 standard deviations below the mean, we can use the formula for calculating a z-score:
\[ z = \frac{(X - \mu)}{\sigma} \]
where:
- \( z \) is the z-score,
- \( X \) is the data point,
- \( \mu \) is the mean,
- \( \sigma \) is the standard deviation.
Given:
- Mean (\( \mu \)) = 254
- Standard Deviation (\( \sigma \)) = 12
A z-score of -2 (indicating 2 standard deviations below the mean) can be calculated as follows:
\[ -2 = \frac{(X - 254)}{12} \]
Multiplying both sides by 12 gives:
\[ -24 = X - 254 \]
Adding 254 to both sides, we find:
\[ X = 254 - 24 = 230 \]
So, the data point that results in a z-score of -2 (2 standard deviations below the mean) is 230.