Question

Given the mean of a data set is 254 and has a standard deviation of 12, which of the following data points would result in a z-score that indicates the data point is 2 standard deviations below the mean? (1 point) 278 230 24 19.167

Answer 1

To find the data point that results in a z-score indicating that it is 2 standard deviations below the mean, we can use the formula for the 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 that:
- mean (\( \mu \)) = 254,
- standard deviation (\( \sigma \)) = 12,
- a z-score of -2 (which indicates 2 standard deviations below the mean).

We can set up the equation:

\[
-2 = \frac{(X - 254)}{12}
\]

Now, solve for \( X \):

1. Multiply both sides by 12:

\[
-24 = X - 254
\]

2. Add 254 to both sides:

\[
X = 254 - 24
\]

\[
X = 230
\]

So, the data point 230 results in a z-score that is 2 standard deviations below the mean.

Therefore, the correct answer is **230**.