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 the85thpercentile?(b) What water footprint represents the27thpercentile?(c) What water footprint represents thethirdquartile? 1.579.57-6.43Mega gallonsμ=1.57 Mgalσ=2.94 Mgalx A normal curve labeled mu = 1.57 mega gallons and sigma = 2.94 mega gallons is over a horizontal x-axis labeled Mega gallons from negative 6.43 to 9.57 in increments of 4 and is centered on 1.57. (a) The water footprint that represents the85thpercentile isenter your response hereMgal.

Solution: Let X be the per capita water footprint (in mega gallons) in a certain country for a recent year can be approximated by a normal distribution with mean 1.57 Mgal and standard deviation 2.94 Mgal. X~N(μ= 1.57, σ= 2.94) a) To find the 85th percentile of water footprint . P(X<a)= 0.85P(X>a)=1-0.85= 0.15PX-μσ>a-μσ= 0.15P(Z>z)= 0.15 < 0.5 z is Positive z = 1.0364 From Excel z= 1.0364=a-μσa =μ+1.0364×σa= 1.57+1.0364×2.94a =1.57+3.047016a= 4.617016a= 4.6170 The water footprint that represents th 85th percentile is 4.6170Mgal b) To find the 27th percentile of water footprint . P(X<a)= 0.27P(X>a)=1-0.27= 0.73PX-μσ>a-μσ= 0.73P(Z>z)= 0.73 >0.5 z is negative z = -0.6128 From Excel z= -0.6128=a-μσa =μ-0.6128×σa= 1.57-0.6128×2.94a =1.57- 1.801632a= -0.231632a= -0.2316 The water footprint that represents th 27th percentile is -0.2316 Mgal c) To find the 3rd quartile of water footprint . P(X<Q3)= 0.75P(X>Q3)=1-0.75= 0.25PX-μσ>Q3-μσ= 0.25P(Z>z)= 0.25 < 0.5 z is positive z = 0.6745 From Excel z= 0.6745=Q3-μσQ3=μ+0.6745×σQ3= 1.57+0.6745×2.94Q3 =1.57+1.98303Q3= 3.55303Q3= 3.5530 The water footprint that represents 3rd quartile is 3.5530 Mgal

Answered: A water footprint is a measure of the…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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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

85th

percentile?

(b) What water footprint represents the

27th

percentile?

third

quartile?

1.579.57-6.43Mega gallons

μ=1.57 Mgal

σ=2.94 Mgal

A normal curve labeled mu = 1.57 mega gallons and sigma = 2.94 mega gallons is over a horizontal x-axis labeled Mega gallons from negative 6.43 to 9.57 in increments of 4 and is centered on 1.57.

(a) The water footprint that represents the

85th

percentile is

enter your response here

Mgal.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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