Question

The American Water Works Association reports that the per capita water use in a single-family home is 76 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-seven owners responded, and the sample mean water use per day was 71 gallons with a standard deviation of 9.5 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

Answer 1

Yes, that is enough statistical evidence to suggest to conclude that residents of Legacy Ranch use less water on average

Explanation:

The data given by the American Water Works Association are;

The per capita water use in a single-family home, μ = 76 gallons per day

The number of responders in the sample, n = 27

The sample mean water used by the sample,
`\overline x` = 71 gallons

The standard deviation, s = 9.5 gallons per day

The level of significance, α = 0.01

The test statistics is given as follows;

The null hypothesis, H₀; μ ≥ 76 gallons

The alternative hypothesis, Hₐ; μ < 76 gallons

`t=\frac{\bar{x}-\mu }{(s)/(√(n))}`

Therefore, we have;

`t=(71-76 )/((9.5)/(√(27))) \approx -2.734817`

At n - 1 = 27 - 1 = 26 degrees of freedom, the critical-t = 2.5

From the test statistic, of t ≈ -2.734817, the p-value < 0.01, and from a graphing calculator, the p-value is given as Prob = 0.005544418

Therefore, given that the p-value is less the significant level, and less than 0.01, is therefore statistically significant, and therefore, there is sufficient statistical evidence against the null hypothesis, therefore we reject the null hypothesis, and there is enough statistical evidence to suggest to conclude that residents of Legacy Ranch use less water on average.