According to a government agency the average daily water con

According to a government agency, the average daily water consumption for a household of four people in a particular country is at least 231 gallons. Suppose another agency plans to test this claim using alpha level equal to 0.02 and a random sample of 200 households with four people. Complete parts a and b.

A. State the appropriate null and alternative hypotheses.

B. Calculate the probability of committing a type II error if the true population mean is 225 gallons. Assume the population standard deviation is known to be 41 gallons.

(please explain steps with out a calculator and how to find answers on standard distribution table or z table)

Solution

Test Used: Z-Test For Single Mean
Set Up Hypothesis
Null, the average daily water consumption for a household H0: U>231
Alternate,the average daily water consumption for a household is less than 231, H1: U<231
Test Statistic
Population Mean(U)=231
Given That X(Mean)=225
Standard Deviation(S.D)=41
Number (n)=200
we use Test Statistic (Z) = x-U/(s.d/Sqrt(n))
Zo=225-231/(41/Sqrt(200)
Zo =-2.0696
| Zo | =2.0696
Critical Value
The Value of |Z | at LOS 0.02% is 2.05
We got |Zo| =2.0696 & | Z | =2.05
Make Decision
Hence Value of | Zo | > | Z | and Here we Reject Ho
P-Value : Left Tail - Ha : ( P < -2.0696 ) = 0.0192
Hence Value of P0.02 > 0.0192, Here we Reject Ho

TYPE I ERROR IS OCCURED . WE have evidence that the average daily water consumption for a household is less than 231