To calculate the [tex]\(z\)[/tex]-score for a given score on the dental anxiety scale, follow these steps:
1. Understand the problem:
- We have scores that are normally distributed with a mean ([tex]\(\mu\)[/tex]) of 10 and a standard deviation ([tex]\(\sigma\)[/tex]) of 4.
- We need to find the [tex]\(z\)[/tex]-score for the score of 16.
2. Recall the formula for the [tex]\(z\)[/tex]-score:
The [tex]\(z\)[/tex]-score is given by the formula:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where:
- [tex]\(X\)[/tex] is the value of the score.
- [tex]\(\mu\)[/tex] is the mean.
- [tex]\(\sigma\)[/tex] is the standard deviation.
3. Substitute the given values into the formula:
- [tex]\(X = 16\)[/tex]
- [tex]\(\mu = 10\)[/tex]
- [tex]\(\sigma = 4\)[/tex]
Plug these values into the formula:
[tex]\[
z = \frac{16 - 10}{4}
\][/tex]
4. Perform the arithmetic operations:
- First, subtract the mean from the score:
[tex]\[
16 - 10 = 6
\][/tex]
- Then, divide this result by the standard deviation:
[tex]\[
\frac{6}{4} = 1.5
\][/tex]
Therefore, the [tex]\(z\)[/tex]-score for a score of 16 on this dental anxiety scale is [tex]\(1.5\)[/tex].
So, [tex]\(z_{16} = 1.5\)[/tex].
