The first thing we have to do is transform the value of the man's height from feet to inches:
[tex]\begin{gathered} 1feet\to12inches \\ 6feet\to72inches \\ 6feet+3inches\to75inches \end{gathered}[/tex]
a) man with 6 feet 3 inches
[tex]\begin{gathered} z_{score}=\frac{(x-\mu)}{\sigma} \\ \mu=69.1 \\ \sigma=2.65 \\ x=75 \\ z_{score}=\frac{(75-69.1)}{2.65} \\ z_{score}=2.23 \end{gathered}[/tex]
b) Using the calculated value of the previous z score, we search the z table to find the percentage of men smaller than that height.
[tex]P(z<2.23)=0.9871[/tex]
So the percentage is 98.81%
c) A woman with 5 feet 11 inches
[tex]5feet+11inches\to71inches[/tex][tex]\begin{gathered} \mu=64.7 \\ \sigma=2.51 \\ x=71 \\ z_{score}=\frac{(x-\mu)}{\sigma} \\ z_{score}=\frac{(71-64.7)}{2.51} \\ z_{score}=2.51 \end{gathered}[/tex]
d) Using the calculated value of the previous z score, we search the z table to find the percentage of women taller than that height.
[tex]\begin{gathered} P(z>2.51)=1-P(z\le2.51) \\ P(z>2.51)=1-0.9940 \\ P(z>2.51)=0.006 \end{gathered}[/tex]
[tex]P(z<2.26)=0.9881[/tex]
So the percentage is 0.6%
