Question

a water footprint is a measure of the appropriation of fresh water. the per capita water footprint (in mega gallons) in a certain country for a recent year can be approximated by a normal distribution, as shown in the figure.
(a) what water footprint represents the 80th percentile?
(b) what water footprint represents the 29th percentile?
(c) what water footprint represents the third quartile?
(a) the water footprint that represents the 80th percentile is □ mgal.
(round to two decimal places as needed.)

Explanation:

Step1: Recall the z - score formula

For a normal distribution $X\sim N(\mu,\sigma^{2})$, the z - score is $z=\frac{x - \mu}{\sigma}$, and we need to find $x$ given a percentile. We first find the z - score corresponding to the given percentile from the standard normal distribution table (z - table), and then solve for $x$. Let $\mu = 1.68$ Mgal and $\sigma=2.2$ Mgal.

Step2: Find the z - score for the 80th percentile

Looking up the value in the standard normal distribution table, the z - score $z$ corresponding to a cumulative probability of $0.80$ is approximately $z = 0.84$.

Step3: Solve for $x$ (water footprint)

Using the z - score formula $z=\frac{x - \mu}{\sigma}$, we can re - arrange it to $x=\mu+z\sigma$. Substituting $\mu = 1.68$, $z = 0.84$, and $\sigma = 2.2$, we get $x=1.68+0.84\times2.2=1.68 + 1.848=3.528\approx3.53$ Mgal.

Step4: Find the z - score for the 29th percentile

The z - score $z$ corresponding to a cumulative probability of $0.29$ from the z - table is approximately $z=- 0.55$.

Step5: Solve for $x$ for the 29th percentile

Using $x=\mu+z\sigma$, substituting $\mu = 1.68$, $z=-0.55$, and $\sigma = 2.2$, we have $x=1.68+( - 0.55)\times2.2=1.68-1.21 = 0.47$ Mgal.

Step6: Recall the definition of the third quartile

The third quartile $Q_{3}$ represents the 75th percentile. The z - score $z$ corresponding to a cumulative probability of $0.75$ from the z - table is approximately $z = 0.67$.

Step7: Solve for $x$ for the third quartile

Using $x=\mu+z\sigma$, substituting $\mu = 1.68$, $z = 0.67$, and $\sigma = 2.2$, we get $x=1.68+0.67\times2.2=1.68 + 1.474=3.154\approx3.15$ Mgal.

Answer:

(a) $3.53$ Mgal
(b) $0.47$ Mgal
(c) $3.15$ Mgal