The American Water Works Association reports that the per ca

The American Water Works Association reports that the per capita water use in a single-family home is 80 gallons per day. Legacy Ranch is a relatively new housing development. The builders installed more efficient water fixtures, such as low-flush toilets, and subsequently conducted a survey of the residences. Twenty-six owners responded, and the sample mean water use per day was 78 gallons with a standard deviation of 7.2 gallons per day.

At the 0.01 level of significance, is that enough evidence to conclude that residents of Legacy Ranch use less water on average?

What is the decision rule? (Negative amount should be indicated by a minus sign. Round your answer to 3 decimal places.)

The value of the test statistic is . (Negative amount should be indicated by a minus sign. Round your answer to 2 decimal places.)

Solution

Given n = 26 µ = 80 x = 78 s = 7.2

The test statistic

   t = (x- µ)/s²/(n-1) t_n-1

the null hypothesis is given by

H₀ : the residents of legancy ranch use less water on average i.e., H₀ : µ 80

Against the alternative hypothesis

H₁ : the residents of legancy ranch use more water on average i.e., H₁ : µ < 80

The test statistic

   t = (x- µ)/s²/(n-1) t_n-1

t = (78- 80)/(7.2)²/(26-1) t_26-1

   t_cal = -1.3889 t₂₅

t_tab = 2.485 at 0.01 level of significance

since t_cal < t_tab we accept the null hypothesis

therefore the residents of legancy ranch use less water on average