Questions: 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 88th percentile? (b) What water footprint represents the 26th percentile? (c) What water footprint represents the third quartile?

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 88th percentile? (b) What water footprint represents the 26th percentile? (c) What water footprint represents the third quartile?
Transcript text: 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 88th percentile? (b) What water footprint represents the 26th percentile? (c) What water footprint represents the third quartile?
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Solution

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Solution Steps

Step 1: Find the z-score for the 26th percentile

Using a z-table or calculator, the z-score corresponding to the 26th percentile (0.26) is approximately -0.64.

Step 2: Calculate the water footprint for the 26th percentile

We are given that the mean (μ) is 1.73 Mgal and the standard deviation (σ) is 2.8 Mgal. We use the formula: x = μ + zσ

x = 1.73 + (-0.64)(2.8) x = 1.73 - 1.792 x ≈ -0.06

Step 3: Find the z-score for the third quartile (75th percentile)

The z-score for the 75th percentile is approximately 0.67.

Step 4: Calculate the water footprint for the third quartile

x = μ + zσ x = 1.73 + (0.67 * 2.8) x = 1.73 + 1.876 x ≈ 3.61

Final Answer:

The water footprint that represents the 26th percentile is approximately -0.06 Mgal. The water footprint that represents the third quartile is approximately 3.61 Mgal.

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