Answer to Question #109633 in Statistics and Probability for andre

Question

In 2014, the per capita consumption of bottled water in Spain
was recorded to be 32.1 gallons. (Data obtained from http://bit
.ly/1pyjHzQ.) Assume that the per capita consumption of bottled
water in Spain is approximately normally distributed with a mean
of 32.1 gallons and a standard deviation of 8 gallons.
a. What is the probability that a person in Spain consumed more
than 31 gallons of bottled water in 2014?
b. What is the probability a person in Spain consumed between 8
and 21 gallons of bottled water in 2014?
c. What is the probability that a person in Spain consumed less
than 8 gallons of bottled water in 2014?
d. Ninety percent of the people in Spain consumed less than how
many gallons of bottled water?

Answer 1

Let x denote the number of gallons of water taken. X~N(32.1, 8²)

a) P(X>31)

Z=$\frac {x- \mu} {\sigma}$

Z=$\frac {31—32.1}{8}$

=-0.1375

P(z>-0.14) from the standard normal table is 0.55567

=0.55567

b)P(8<X<21)

=P(x<21)-p(x<8)

P(x<21)

Z1=$\frac {21-32.1}{8}$ =-1.3875

P(Z1<-1.39) is 0.08226 from the z tables

P(x<8)

Z2=$\frac {8-32.1}{8}$

=-3.0125

P(Z2<-3.01) is 0.00131 from the standard normal tables.

P(8<X<21)= 0.08226-0.00131

=0.08095

c) P(x<8)

Z=$\frac {8-32.1}{8}$

=-3.0125

P(z<-3.01) is 0.00131 from the z tables

=0.00131

d) P(X<x ) = 0.90

$\Phi ^{-1}(0.9)=1.2816$ from the z tables

1.2816=$\frac {x-32.1}{8}$

x=42.35 gallons of bottled water