Q:

Water use in the summer is normally distributed with a mean of 311.4 million gallons per day and a standard deviation of 40 million gallons per day. City reservoirs have a combined storage capacity of 350 million gallons. The probability that a day requires more water than is stored in city reservoirs is P(X > 350)= 1 - P (Z < b). What is the value of b? Please report your answer in 3 decimal places.

Accepted Solution

A:
Answer: 0.965Step-by-step explanation:Given : Water use in the summer is normally distributed with [tex]\mu=311.4\text{ million gallons per day}[/tex][tex]\sigma=40 \text{ million gallons per day}[/tex]Let X be the random variable that represents the quantity of water required on a particular day.Z-score : [tex]\dfrac{x-\mu}{\sigma}[/tex][tex]\dfrac{350-311.4}{40}=0.965[/tex]Now, the probability that a day requires more water than is stored in city reservoirs is given by:-[tex]P(x>350)=P(z>0.965)=1-P(z<0.965)[/tex]We can see that on comparing the above value to the given P(X > 350)= 1 - P(Z < b) , we get the value of b is 0.965.