Answer
a. $0.9787$
b. $21.08$ to $24.22$
c. $0.9998$; No
Work Step by Step
a) We first find the two $z$-scores:
$z=\frac{25−22.65}{0.8}=2.94$
$z=\frac{21−22.65}{0.8}=−2.06$
Using the table of z-scores, we find:
$0.9984−0.0197=0.9787$
b) Using the table of negative $z$-scores, we can find that this corresponds to a value of $z=±1.96$. Hence:
$max=22.65+(0.8)(1.96)=24.22$
$min=22.65+(0.8)(−1.96)=21.08$
c) Using $z$-scores, we can find that the probability is $0.9998$. However, because this represents the mean of women and not the general population, it does not follow that the majority of women will be able to fit in these hats.
