Answer :
To determine which mileage corresponds to a [tex]$z$[/tex]-score of -1.5, we need to use the formula that relates the [tex]$z$[/tex]-score to a specific data point within a normally distributed set of data. The formula for the [tex]$z$[/tex]-score is given by:
[tex]\[ z = \frac{x - \bar{x}}{s} \][/tex]
where:
- [tex]\( x \)[/tex] is the value we are trying to find (in this case, the mileage corresponding to a [tex]$z$[/tex]-score of -1.5),
- [tex]\( \bar{x} \)[/tex] is the mean of the data (which is 29 miles),
- [tex]\( s \)[/tex] is the standard deviation of the data (which is 3.6 miles).
Let's rearrange the formula to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \bar{x} + z \cdot s \][/tex]
Now, plug in the given values:
- [tex]\(\bar{x} = 29\)[/tex]
- [tex]\(z = -1.5\)[/tex]
- [tex]\(s = 3.6\)[/tex]
Substitute these values into the formula:
[tex]\[ x = 29 + (-1.5) \cdot 3.6 \][/tex]
Calculate the product of [tex]\(z\)[/tex] and [tex]\(s\)[/tex]:
[tex]\[ -1.5 \cdot 3.6 = -5.4 \][/tex]
Now add this result to the mean:
[tex]\[ x = 29 - 5.4 \][/tex]
[tex]\[ x = 23.6 \][/tex]
Therefore, the mileage that corresponds to a [tex]$z$[/tex]-score of -1.5 is 23.6 miles. Among the given options, the closest mileage is:
24 miles
[tex]\[ z = \frac{x - \bar{x}}{s} \][/tex]
where:
- [tex]\( x \)[/tex] is the value we are trying to find (in this case, the mileage corresponding to a [tex]$z$[/tex]-score of -1.5),
- [tex]\( \bar{x} \)[/tex] is the mean of the data (which is 29 miles),
- [tex]\( s \)[/tex] is the standard deviation of the data (which is 3.6 miles).
Let's rearrange the formula to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \bar{x} + z \cdot s \][/tex]
Now, plug in the given values:
- [tex]\(\bar{x} = 29\)[/tex]
- [tex]\(z = -1.5\)[/tex]
- [tex]\(s = 3.6\)[/tex]
Substitute these values into the formula:
[tex]\[ x = 29 + (-1.5) \cdot 3.6 \][/tex]
Calculate the product of [tex]\(z\)[/tex] and [tex]\(s\)[/tex]:
[tex]\[ -1.5 \cdot 3.6 = -5.4 \][/tex]
Now add this result to the mean:
[tex]\[ x = 29 - 5.4 \][/tex]
[tex]\[ x = 23.6 \][/tex]
Therefore, the mileage that corresponds to a [tex]$z$[/tex]-score of -1.5 is 23.6 miles. Among the given options, the closest mileage is:
24 miles
