Question

In Jefferson county, the mean water usage per household for the month of January 2018 was .060 acre-feet (an acre-foot is the amount of water necessary to cover one acre to a depth of one foot.) This figure is based on the usage of all households in the county during that month.  In January 2019, a random sample of 50 homes was selected, and water usage was recorded for each home in the sample. The mean water usage per household in the sample was .054 acre-feet with a standard deviation (s) of .016 acre-feet.

The county would like to know whether the data support the claim that mean January water usage has declined. Conduct an appropriate statistical hypothesis test using a 0.01 significance level. Find the associated the P-value to four decimal places

Answer 1

To Test :-
H0 :- µ = 0.060
H1 :- µ < 0.060

Test Statistic :-
t = ( X̅ - µ ) / (S / √(n) )
t = ( 0.054 - 0.06 ) / ( 0.016 / √(50) )
t = -2.6517

Test Criteria :-
Reject null hypothesis if t < -t(α, n-1)
t(α, n-1) = t(0.01 , 50-1) = 2.405
t < - t(α, n-1) = -2.6517 < - 2.405
Result :- Reject null hypothesis

Decision based on P value
P - value = P ( t > 2.6517 ) = 0.0054
Reject null hypothesis if P value < α = 0.01 level of significance
P - value = 0.0054 < 0.01 ,hence we reject null hypothesis
Conclusion :- Reject null hypothesis

There is sufficient evidence to support the claim that mean January water usage has declined.