(a)
[tex]\begin{gathered} z=\frac{x-\mu}{\sigma} \\ x=12.5,\mu=19.7,\sigma=3.6 \\ z=\frac{12.5-19.7}{3.6} \\ z=-\frac{7.2}{3.6} \\ z=-2 \end{gathered}[/tex][tex]\begin{gathered} x=26.9 \\ z=\frac{26.9-19.7}{3.6} \\ z=\frac{7.2}{3.6} \\ z=2 \end{gathered}[/tex]
The percentage of showers that used between 12.5 and 26.9 gallons is;
[tex]\begin{gathered} P(-2
(b) The percentage of showers that used
more than 26.9 gallons is;[tex]\begin{gathered} P(x>2)=0.02275 \\ P(x>2)=2.275\text{ \%} \\ P(x>2)\cong2\text{ \%} \end{gathered}[/tex]
(c)
[tex]\begin{gathered} x=16.1 \\ z=\frac{16.1-19.7}{3.6} \\ z=-\frac{3.6}{3.6} \\ z=-1 \end{gathered}[/tex]
The percentage of showers that used less than 16.1 gallons is;
[tex]\begin{gathered} P(x<-1)=0.15866 \\ P(x<-1)=15.866\text{ \%} \\ P(x<-1)\cong16\text{ \%} \end{gathered}[/tex]
(d)
[tex]\begin{gathered} x=30.5 \\ z=\frac{30.5-19.7}{3.6} \\ z=\frac{10.8}{3.6} \\ z=3 \end{gathered}[/tex]
The percentage of showers that used between 16.1 and 30.5 gallons is;
[tex]\begin{gathered} P(-1
