Question

1) Psychology students at Wittenberg University completed the Dental Anxiety Scale questionnaire. 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 with \mu = 11 and \sigma = 3.5.

a) Find the probability that someone scores between a 10 and a 15 on the Dental Anxiety Scale.
b) Find the probability that someone scores above a 17 on the Dental Anxiety Scale.
2) Ecological Applications published a study on the development of forests following wildfires in the Pacific Northwest. One variable of interest to the researcher was tree diameter at breast height 110 years after the fire. The population of Douglas fir trees was shown to have an approximately normal diameter distribution, with \mu = 50 centimeters and \sigma = 12 cm. Find the diameter, d, such that 30% of the Douglas fir trees in the population have diameters that exceed d.

Answer 1

1) 0.487,0.0436

2) 56.3 cm

Explanation:

Let X be the scores ranging from 0 to 20 of the Psychology students at Wittenberg University completed the Dental Anxiety Scale questionnaire.

X is N(11,3.5)

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

=
`P(10<x<15) = P((10-11)/(3.5) <z<(15-11)/(3.5))\\=P(-0.29<z<1.14)\\=0.3729+0.1141\\=0.4870`

b)
`P(X>17) = P(Z>1.71)\\= 0.5-0.4564\\=0.0436`

c) Y is N(50, 20) where Y is tree diameter at breast height 110 years after the fire.

P(X>d) = 0.30

Corresponding Z value = 0.525

`X= 50+12(0.525)\\=56.3`

d = 56.3 cm