Question

Psychology students at Wittenberg University completed the Dental Anxiety Scale questionnaire (Psychological Reports, August 1997). Scores on the scale range from 0 (no anxiety) to 20 (extreme anxiety). The mean score was 11 and the standard deviation was 3.5. Assume that the distribution of all scores on the Dental Anxiety Scale is normal with μ = 11 and σ = 3.5. (a) Suppose you score a 16 on the Dental Anxiety Scale. Find the z-value for this score. (b) Find the probability that someone scores between a 10 and a 15 on the Dental Anxiety Scale. (c) Find the probability that someone scores above a 17 on the Dental Anxiety Scale

Answer 1

Answer:

a) $Z = 1.43$

b) There is a 48.696% probability that someone scores between a 10 and a 15 on the Dental Anxiety Scale is

Step-by-step explanation:

Normal model problems can be solved by the zscore formula.

On a normaly distributed set with mean $\mu$ and standard deviation $\sigma$, the z-score of a value X is given by:

$Z = \frac{X - \mu}{\sigma}$

Each z-score value has an equivalent p-value, that represents the percentile that the value X is.

In our problem, the mean score was 11 and the standard deviation was 3.5.

So, $\mu = 11$, $\sigma = 3.5$.

(a) Suppose you score a 16 on the Dental Anxiety Scale. Find the z-value for this score.

What is the value of Z when $X = 16$?

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{16 - 11}{3.5} = 1.43$

(b) Find the probability that someone scores between a 10 and a 15 on the Dental Anxiety Scale.

We have to find the percentiles of both of these scores. This means that we have to find Z when $X = 10$ and $X = 15$. The probability that someone scores between a 10 and a 15 is the difference between the pvalues of the z-value of X = 10 and X = 15.

When $X = 10$

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{10 - 11}{3.5} = -0.29$

Looking at the z score table, we find that the pvlaue of $Z = -0.29$ is 0.3859.

When $X = 15$

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{15 - 11}{3.5} = 1.14$

Looking at the z score table, we find that the pvlaue of $Z = 1.14$ is 0.87286.

So, the probability that someone scores between a 10 and a 15 on the Dental Anxiety Scale is

0.87286 - 0.3859 = 0.48696 = 48.696%

(c) Find the probability that someone scores above a 17 on the Dental Anxiety Scale

This probability is 100% minus the pvalue of the zvalue when $X = 17$

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{17 - 11}{3.5} = 1.71$

When $Z = 1.71$, the pvalue is 0.95637. This means that there is a 95.637% probability that someone scores BELOW 17 on the dental anxiente scale.

100 - 95.637 = 4.363%

There is 4.363% probability that someone scores above a 17 on the Dental Anxiety Scale