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Answer :
To find the [tex]$z$[/tex]-score for a given value [tex]$X = 85$[/tex] in a population with a mean [tex]$\mu = 80$[/tex] and a standard deviation [tex]$\sigma = 10$[/tex], you can follow these steps:
1. Understand the formula for the [tex]$z$[/tex]-score:
The [tex]$z$[/tex]-score formula is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
Where:
- [tex]\( z \)[/tex] is the [tex]$z$[/tex]-score we're trying to find.
- [tex]\( X \)[/tex] is the given value from the population.
- [tex]\( \mu \)[/tex] is the mean of the population.
- [tex]\( \sigma \)[/tex] is the standard deviation of the population.
2. Substitute the given values into the formula:
In this problem, the given values are:
- [tex]\( X = 85 \)[/tex]
- [tex]\( \mu = 80 \)[/tex]
- [tex]\( \sigma = 10 \)[/tex]
Substituting these values into the formula:
[tex]\[
z = \frac{85 - 80}{10}
\][/tex]
3. Perform the calculation:
- First, subtract the mean from [tex]\( X \)[/tex]:
[tex]\[
85 - 80 = 5
\][/tex]
- Next, divide the result by the standard deviation:
[tex]\[
\frac{5}{10} = 0.5
\][/tex]
So, the [tex]$z$[/tex]-score corresponding to [tex]$X = 85$[/tex] is [tex]\( +0.50 \)[/tex].
1. Understand the formula for the [tex]$z$[/tex]-score:
The [tex]$z$[/tex]-score formula is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
Where:
- [tex]\( z \)[/tex] is the [tex]$z$[/tex]-score we're trying to find.
- [tex]\( X \)[/tex] is the given value from the population.
- [tex]\( \mu \)[/tex] is the mean of the population.
- [tex]\( \sigma \)[/tex] is the standard deviation of the population.
2. Substitute the given values into the formula:
In this problem, the given values are:
- [tex]\( X = 85 \)[/tex]
- [tex]\( \mu = 80 \)[/tex]
- [tex]\( \sigma = 10 \)[/tex]
Substituting these values into the formula:
[tex]\[
z = \frac{85 - 80}{10}
\][/tex]
3. Perform the calculation:
- First, subtract the mean from [tex]\( X \)[/tex]:
[tex]\[
85 - 80 = 5
\][/tex]
- Next, divide the result by the standard deviation:
[tex]\[
\frac{5}{10} = 0.5
\][/tex]
So, the [tex]$z$[/tex]-score corresponding to [tex]$X = 85$[/tex] is [tex]\( +0.50 \)[/tex].
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