b. The probability is approximately
0.4768
σ
x
x
= σ ÷ √(n)
= 2.8 ÷ √(15)
≈ 0.7229568913
z
low
= (62.0 – 62.2) ÷ 0.7229568913
≈ -0.2766416676
≈ -0.28
z
up
≈ (63.0 – 62.2) ÷ 0.47229568913
≈ 1.10656667
≈ 1.11
The probability is the area between z = -0.28 and z = 1.11 under the standard normal distribution.
0.8665 – 0.3897 = 0.4768
.
43

Women have head circumferences that are normally distributed with a mean given by
µ
= 24.27 in., and a standard
deviation given by
σ
= 0.8 in.
a.
If a hat company produces women's hats so that they fit head circumferences between 23.6 in. and 24.6
in., what is the probability that a randomly selected woman will be able to fit into one of these hats?
b.
If the company wants to produce hats to fit all women except for those with the smallest 1.25% and the
largest 1.25% head circumferences, what head circumferences should be accommodated?

d.
If this probability is high, does it suggest that an order of 12 hats will very likely fit each of 12 randomly
selected women? Why or why not? (Assume that the hat company produces women's hats so that they fit
head circumferences between 23.6 in. and 24.6 in.)
a.
The probability is
0.4586
.

max = (2.24)(0.8) + 24.27 = 26.062
c.
The probability is
0.9217
.
σ
x
x
= σ ÷ √(n)
= 0.8 ÷ √(12)
≈ 0.2309401077
z
low
= (23.6 – 24.27) ÷ 0.2309401077
≈ -2.90
z
up
≈ (24.6 – 24.27) ÷ 0.2309401077
≈ 1.43
The probability is the area between z = -2.90 and z = 1.43 under the standard normal distribution.
0.9236 – 0.0019 = 0.9217
d.
No, the hats must fit individual women, not the mean from 12 women. If all hats are made to fit head circumferences
between 23.6 in. and 24.6 in., the hats won't fit about half of those women.
44
A ski gondola carries skiers to the top of a mountain. It bears a plaque stating that the maximum capacity is 12 people or
1920 lb. That capacity will be exceeded if 12 people have weights with a mean greater than (1920 lb)/12 = 160 lb. Assume
that weights of passengers are normally distributed with a mean of 177.5 lb and a standard deviation of 40.4 lb.
a.
Find the probability that if an individual passenger is randomly selected, their weight will be greater than
160 lb.
b.
Find the probability that 12 randomly selected passengers will have a mean weight that is greater than 160
lb (so that their total weight is greater than the gondola maximum capacity of 1920 lb)