Answer to Question #109625 in Statistics and Probability for Dewi

Question

6.7 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.

Answer 1

Let $X=$ the number of consumed of bottled water: $X\sim N(\mu, \sigma^2).$

Then $Z=\dfrac{X-\mu}{\sigma}\sim N(0,1)$

Given: $\mu=32.1, \sigma=8.$

a.The probability that someone consumed more than 32 gallons of bottled water is

$$P(X>32)=1-P(X\leq 32)=$$

$$=1-P(Z\leq {32-32.1 \over8})=1-P(Z\leq -0.0125)\approx$$

$$\approx0.504987$$

b. The probability that someone consumed between 25 and 35 gallons of bottled water is

$$P(25<X<35)=P(X<35)-P(X<25)=$$

$$=P(Z<{35-32.1 \over8})-P(Z< {25-32.1 \over8})=$$

$$=P(Z<0.3625)-P(Z<-0.8875)\approx$$

$$\approx0.6415108-0.1874049\approx0.454106$$

c. The probability that someone consumed less than 25 gallons of bottled water is

$$P(X<25)=P(Z< {25-32.1 \over8})=$$

$$P(Z<-0.8875)\approx0.187405$$

d. 99 % of people consumed less than how many gallons of bottled water?

$$P(X<X^*)=P(Z<{X^*-32.1 \over8})=0.99$$

$${X^*-32.1 \over8}\approx2.32635$$

$$X^*\approx50.7108\approx51$$