According to data from the Environmental Protection Agency t

According to data from the Environmental Protection Agency, the average daily water consumption for a household of four people in the United States is approximately at least 243 gallons. (Source: http://www.catskillcenter.org/programs/csp/H20/Lesson3/house3.htm) Suppose a state agency plans to test this claim using an alpha level equal to 0.05 and a random sample of 100 households with four people.

Calculate the probability of committing a Type II error if the true population mean is 230 gallons. Assume that the population standard deviation is known to be 40 gallons.

Solution

Test Used: Z-Test For Single Mean
Set Up Hypothesis
Null, H0: U>=243
Alternate, H1: U<243 ( left tailed - test)
Test Statistic
Population Mean(U)=243
Standard Deviation(S.D)=40
Number (n)=100
we use Test Statistic (Z) = x-U/(s.d/Sqrt(n))
Zo=x-243/(40/Sqrt(100)
Zo =x-243/(40/10)
Zo =x-243/4
Critical Value
The Value of Z at LOS 0.05% is -1.645

=> x-243/4 = -1.645
=> x = 236.42

The true population is given as 230
P( Z <=x-U/(s.d/Sqrt(n)) = P( Z <= 236.42-230/(40/Sqrt(100)
                       = P( Z <= 1.605)
                       = 0.9458  

                      
The Probability of Type II Error (Beta) = 1 - 0.9458 = 0.0542 ~ Near by Value 0.0537 [ANSWER]