Answer:
a) ![Z = 1.43](https://tex.z-dn.net/?f=Z%20%3D%201.43)
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
and standard deviation
, the z-score of a value X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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,
,
.
(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
?
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{16 - 11}{3.5} = 1.43](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B16%20-%2011%7D%7B3.5%7D%20%3D%201.43)
(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
and
. 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](https://tex.z-dn.net/?f=X%20%3D%2010)
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{10 - 11}{3.5} = -0.29](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B10%20-%2011%7D%7B3.5%7D%20%3D%20-0.29)
Looking at the z score table, we find that the pvlaue of
is 0.3859.
When ![X = 15](https://tex.z-dn.net/?f=X%20%3D%2015)
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{15 - 11}{3.5} = 1.14](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B15%20-%2011%7D%7B3.5%7D%20%3D%201.14)
Looking at the z score table, we find that the pvlaue of
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](https://tex.z-dn.net/?f=X%20%3D%2017)
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{17 - 11}{3.5} = 1.71](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B17%20-%2011%7D%7B3.5%7D%20%3D%201.71)
When
, 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