To find the age of a voter corresponding to a [tex]$z$[/tex]-score of [tex]$1.14$[/tex], we can use the formula for converting a [tex]$z$[/tex]-score to a data point in a normal distribution:
[tex]\[ x = \mu + z \times \sigma \][/tex]
where:
- [tex]\( \mu \)[/tex] is the mean,
- [tex]\( \sigma \)[/tex] is the standard deviation,
- [tex]\( z \)[/tex] is the [tex]$z$[/tex]-score,
- [tex]\( x \)[/tex] is the data point.
Given:
- Mean ([tex]\( \mu \)[/tex]) = 65,
- Standard Deviation ([tex]\( \sigma \)[/tex]) = 5,
- [tex]$z$[/tex]-score ([tex]$z_x$[/tex]) = 1.14,
we can substitute these values into the equation:
[tex]\[ x = 65 + 1.14 \times 5 \][/tex]
First, compute the product of the [tex]$z$[/tex]-score and the standard deviation:
[tex]\[ 1.14 \times 5 = 5.7 \][/tex]
Next, add this result to the mean:
[tex]\[ x = 65 + 5.7 = 70.7 \][/tex]
Therefore, the age corresponding to the [tex]$z$[/tex]-score of 1.14 is 70.7. When rounding this to the nearest whole number, we get 71.
So, the possible voter age that would give her [tex]$z_x = 1.14$[/tex] is:
[tex]\[ \boxed{71} \][/tex]
