Question

4. Suppose that water usages in American showers are normally distributed, with an average shower using 19.7 gallons, and a standard deviation of 3.6 gallons. Estimate the percentage of showers that used(a) between 12.5 and 26.9 gallons. %(b) more than 26.9 gallons. %(c) less than 16.1 gallons. %(d) between 16.1 and 30.5 gallons. %

Answer 1

(a)

`\begin{gathered} z=(x-\mu)/(\sigma) \\ x=12.5,\mu=19.7,\sigma=3.6 \\ z=(12.5-19.7)/(3.6) \\ z=-(7.2)/(3.6) \\ z=-2 \end{gathered}`
`\begin{gathered} x=26.9 \\ z=(26.9-19.7)/(3.6) \\ z=(7.2)/(3.6) \\ z=2 \end{gathered}`

The percentage of showers that used between 12.5 and 26.9 gallons is;

(c)

`\begin{gathered} x=16.1 \\ z=(16.1-19.7)/(3.6) \\ z=-(3.6)/(3.6) \\ z=-1 \end{gathered}`

The percentage of showers that used less than 16.1 gallons is;

`\begin{gathered} P(x<-1)=0.15866 \\ P(x<-1)=15.866\text{ \%} \\ P(x<-1)\cong16\text{ \%} \end{gathered}`

(d)

`\begin{gathered} x=30.5 \\ z=(30.5-19.7)/(3.6) \\ z=(10.8)/(3.6) \\ z=3 \end{gathered}`

The percentage of showers that used between 16.1 and 30.5 gallons is;

[tex]\begin{gathered} P(-1