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Scores on a dental anxiety scale range from 0 (no anxiety) to 20 (extreme anxiety). The scores are normally distributed with a mean of 10 and a standard deviation of 4.

Find the z-score for a given score of 16.

[tex]\[ z_{16} = \ \square \][/tex] (Type an integer or a decimal.)



Answer :

GinnyAnswer
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].

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