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

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

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 = (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 = (X - \mu)/(\sigma)`

`Z = (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 = (X - \mu)/(\sigma)`

`Z = (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 = (X - \mu)/(\sigma)`

`Z = (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 = (X - \mu)/(\sigma)`

`Z = (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