A Find the probability that a randomly chosen household uses

A. Find the probability that a randomly chosen household uses more than 23 gallons per day.

B. Find the probability that a randomly chosen household uses between 20 and 25 gallons per day.

C. If the mayor wants to give a tax rebate to the 20% lowest water users, what should the gallons per day cutoff be?

Solution

Mean ( u ) =25
Standard Deviation ( sd )=4
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X > 23) = (23-25)/4
= -2/4 = -0.5
= P ( Z >-0.5) From Standard Normal Table
= 0.6915                  
b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 20) = (20-25)/4
= -5/4 = -1.25
= P ( Z <-1.25) From Standard Normal Table
= 0.10565
P(X < 25) = (25-25)/4
= 0/4 = 0
= P ( Z <0) From Standard Normal Table
= 0.5
P(20 < X < 25) = 0.5-0.10565 = 0.3944                  
c)
P ( Z < x ) = 0.2
Value of z to the cumulative probability of 0.2 from normal table is -0.842
P( x-u/s.d < x - 25/4 ) = 0.2
That is, ( x - 25/4 ) = -0.84
--> x = -0.84 * 4 + 25 = 21.632