EXAMPLE 6.17

Water quality testing. The Deely Laboratory is a drinking-water testing and analysis service. One of the common contaminants it tests for is lead. Lead enters drinking water through corrosion of plumbing materials, such as lead pipes, fixtures, and solder. The service knows that their analysis procedure is unbiased but not perfectly precise, so the laboratory analyzes each water sample three times and reports the mean result. The repeated measurements follow a Normal distribution quite closely. The standard deviation of this distribution is a property of the analytic procedure and is known to be σ = 0.25 parts per billion (ppb).

The Deely Laboratory has been asked by a university to evaluate a claim that the drinking water in the Student Union has a lead concentration above the Environmental Protection Agency’s (EPA) action level of 15 ppb. Because the true concentration of the sample is the mean μ of the population of repeated analyses, the hypotheses are

H₀: μ = 15

H_a: μ ≠ 15

We use the two-sided alternative here because there is no prior evidence to substantiate a one-sided alternative. The lab chooses the 1% level of significance, α = 0.01.

Three analyses of one specimen give concentrations

15.84 15.33 15.58

The sample mean of these readings is

The test statistic is

Because the alternative is two-sided, the P-value is

P = 2P(Z ≥ 4.02)

We cannot find this probability in Table A. The largest value of z in that table is 3.49. All that we can say from Table A is that P is less than 2P(Z ≥ 3.49) = 2(1−0.9998) = 0.0004. Software or a calculator could be used to give an accurate value of the P-value. However, because the P-value is clearly less than the lab’s standard of 1%, we reject H₀. Because is larger than 15.00, we can conclude that the true concentration level of lead in this one specimen is higher than the EPA’s action level.