To find the [tex]$z$[/tex]-score of an appliance that stopped working at 64 months, we need to use the [tex]$z$[/tex]-score formula. The [tex]$z$[/tex]-score formula is:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]
where:
- [tex]\( x \)[/tex] is the value we are interested in (the lifespan of the appliance that stopped working),
- [tex]\( \mu \)[/tex] is the mean of the distribution,
- [tex]\( \sigma \)[/tex] is the standard deviation of the distribution.
Given:
- The mean [tex]\(\mu\)[/tex] is 48 months,
- The standard deviation [tex]\(\sigma\)[/tex] is 8 months,
- The value [tex]\( x \)[/tex] is 64 months.
We can plug these values into the formula:
[tex]\[ z = \frac{64 - 48}{8} \][/tex]
First, compute the numerator:
[tex]\[ 64 - 48 = 16 \][/tex]
Then, divide by the standard deviation:
[tex]\[ z = \frac{16}{8} = 2 \][/tex]
Therefore, the [tex]$z$[/tex]-score of an appliance that stopped working at 64 months is:
[tex]\[ 2 \][/tex]
So the correct answer is:
[tex]\[ \boxed{2} \][/tex]
