Sure, let's determine a possible voter age given that the mean age of the voters [tex]$\mu$[/tex] (mu) is 65 years and the standard deviation [tex]$\sigma$[/tex] (sigma) is 5 years. The z-score [tex]$z_x$[/tex] is 1.14.
First, we need to recall the formula for the z-score:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where:
- [tex]\( z \)[/tex] is the z-score,
- [tex]\( X \)[/tex] is the value we need to find (the voter age in this case),
- [tex]\( \mu \)[/tex] is the mean of the ages,
- [tex]\( \sigma \)[/tex] is the standard deviation of the ages.
Rearranging the formula to solve for [tex]\( X \)[/tex], we get:
[tex]\[
X = z \times \sigma + \mu
\][/tex]
Substituting the given values:
[tex]\[
X = 1.14 \times 5 + 65
\][/tex]
Now performing the multiplication and addition:
[tex]\[
X = 5.7 + 65 = 70.7
\][/tex]
Thus, the possible voter age calculated is 70.7 years. When we round 70.7 to the nearest whole number, we obtain:
[tex]\[
\boxed{71}
\][/tex]
Therefore, the possible voter age that would give a z-score of 1.14 is 71.
