1. Assume that the population of heights of female college students is approximately
normally distributed with mean m of 66 inches and standard deviation s of 4.00 inches.
Show all work.
(A) Find the proportion of female college students whose height is greater than 65 inches.
Calculate the z-score: z =
x " µ 65 " 66
=
= "0.25
#
4
The proportion with a height greater than 65 inches will be equal to the area under
the normal curve to the right of z = -0.25.
!
P(x > 65) = P(z > -0.25) = 0.5987
(B) Find the proportion of female college students whose height is no more than 65
inches.
The proportion with a height less than or equal to 65 inches will be equal to the
difference between 1 and the answer from part A.
P(x ≤ 65) = 1 – P(x > 65) = 1 – 0.5987 = 0.4013
2. The diameters of grapefruits in a certain orchard are normally distributed with a mean
of 7.10 inches and a standard deviation of 0.50 inches. Show all work.
(A) What percentage of the grapefruits in this orchard have diameters less than 6.8
inches?
Calculate the z-score: z =
x " µ 6.8 " 7.10
=
= "0.60
#
0.5
P(x < 6.8) = P(z < -0.60) = 0.2743 = 27.43%
!
(B) What percentage of the grapefruits in this orchard are larger than 7.00 inches?
Calculate the z-score: z =
x " µ 7.0 " 7.10
=
= "0.20
#
0.5
P(x > 7.0) = P(z > -0.20) = 0.5793 = 57.93%
!
3. A set of data is normally distributed with a mean of 500 and standard deviation of 100.
· What would be the standard score for a score of 400?
z=
!
x " µ 400 " 500
=
= "1
#
100
· What percentage of scores is between 500 and 400?
We know from the previous answer that a score of 400 corresponds to a z-value
of -1. We also know that if the mean is 500, then a score of 500 corresponds to a
z-value of 0.
So, this question becomes, what is the area between z = -1 and z = 0.
The area to the left of z = -1 is 0.1587.
The area between z = -1 and z = 0 is then 0.5 – 0.1587 = 0.3413
Converted to a percentage, 0.3413 = 34.13%
· What would be the percentile rank for a score of 400?
We know from the previous answer that the z-value for a score of 400 is -1. We
also know that the area under the normal curve to the left of z = -1 is 0.1587.
The percentage of scores below 400 is equal to this area, expressed as a
percentage:
0.1587 = 15.87%
Since 15.86% of the scores are below 400, a score of 400 is at the 16th percentile,
rounding off to the nearest percent.