To determine which statement is true about the distance of 37 miles from the mean, we will follow these steps:
1. Calculate the Z-score for 37 miles:
The Z-score tells us how many standard deviations a data point is from the mean. The formula for the Z-score is:
[tex]\[
z = \frac{y - \bar{x}}{s}
\][/tex]
Where:
- [tex]\( y \)[/tex] is the value we're interested in (37 miles).
- [tex]\( \bar{x} \)[/tex] is the mean (29 miles).
- [tex]\( s \)[/tex] is the standard deviation (3.6 miles).
2. Substitute the values into the formula:
[tex]\[
z_{37} = \frac{37 - 29}{3.6}
\][/tex]
3. Perform the subtraction and division:
[tex]\[
z_{37} = \frac{8}{3.6} = 2.2222222222222223
\][/tex]
So, the Z-score for 37 miles is approximately [tex]\( 2.222 \)[/tex].
4. Determine which interval the Z-score falls into:
- If the Z-score is less than or equal to 1, the value is within 1 standard deviation of the mean.
- If the Z-score is between 1 and 2, the value is between 1 and 2 standard deviations of the mean.
- If the Z-score is between 2 and 3, the value is between 2 and 3 standard deviations of the mean.
- If the Z-score is greater than 3, the value is more than 3 standard deviations from the mean.
Since [tex]\( z_{37} = 2.222 \)[/tex]:
- It is not within 1 standard deviation of the mean.
- It is not between 1 and 2 standard deviations of the mean.
- It is between 2 and 3 standard deviations of the mean.
- It is not more than 3 standard deviations from the mean.
Therefore, the correct statement is:
[tex]\[ z_{37} \text{ is between 2 and 3 standard deviations of the mean.} \][/tex]
