Sure! Let's solve this step by step:
1. Understanding the problem:
We need to find the mileage corresponding to a given z-score. The mean distance ([tex]\(\bar{x}\)[/tex]) that employees live from work is 29 miles, and the standard deviation ([tex]\(\sigma\)[/tex]) is 6 miles. We are given a z-score of -1.5 and need to identify which of the given options (21, 24, 36, and 41 miles) is closest to the actual mileage associated with this z-score.
2. Formula for z-score:
The formula to calculate the mileage from a z-score is:
[tex]\[
X = \bar{x} + z \cdot \sigma
\][/tex]
where [tex]\(X\)[/tex] is the mileage, [tex]\(\bar{x}\)[/tex] is the mean mileage, [tex]\(z\)[/tex] is the z-score, and [tex]\(\sigma\)[/tex] is the standard deviation.
3. Substitute the given values:
[tex]\[
X = 29 + (-1.5) \cdot 6
\][/tex]
4. Calculate the mileage:
[tex]\[
X = 29 - 9 = 20
\][/tex]
The mileage corresponding to a z-score of -1.5 is 20 miles.
5. Identify the closest option:
Comparing the calculated mileage (20 miles) with the given options:
- 21 miles
- 24 miles
- 36 miles
- 41 miles
The closest value to 20 miles among these options is 21 miles.
Therefore, the mileage within a z-score of -1.5 is approximately 21 miles.
