To determine a possible voter age that corresponds to a z-score of [tex]\( z_x = 1.14 \)[/tex] given a mean ([tex]\(\mu\)[/tex]) of 65 and a standard deviation ([tex]\(\sigma\)[/tex]) of 5, follow these steps:
1. Recall the Z-score formula:
[tex]\[
z = \frac{x - \mu}{\sigma}
\][/tex]
where:
- [tex]\( z \)[/tex] is the z-score
- [tex]\( x \)[/tex] is the value for which we are calculating the z-score
- [tex]\( \mu \)[/tex] is the mean
- [tex]\( \sigma \)[/tex] is the standard deviation
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = z \cdot \sigma + \mu
\][/tex]
Here, [tex]\( z_x = 1.14 \)[/tex], [tex]\(\mu = 65\)[/tex], and [tex]\(\sigma = 5 \)[/tex].
3. Substitute the values given:
[tex]\[
x = 1.14 \cdot 5 + 65
\][/tex]
4. Calculate the result:
[tex]\[
x = 5.7 + 65
\][/tex]
[tex]\[
x = 70.7
\][/tex]
5. Round [tex]\( x \)[/tex] to the nearest whole number:
[tex]\[
70.7 \approx 71
\][/tex]
Therefore, a possible voter age that would give a z-score of [tex]\( z_x = 1.14 \)[/tex] is 71.
