18-Env-B5 Industrial & Hazardous Waste Management · December 2014
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Reference texts: Nemerow & Dasgupta, Industrial and Hazardous Waste Treatment, 2nd ed.; Metcalf & Eddy, Wastewater Engineering: Treatment and Resource Recovery, 5th ed.; Davis & Cornwell, Introduction to Environmental Engineering, 6th ed.; LaGrega, Buckingham & Evans, Hazardous Waste Management, 2nd ed.; CCME, Guidelines for the Management of Biomedical Waste in Canada (1992); Canadian Environmental Protection Act (CEPA), 1999; provincial Environmental Protection / Hazardous Waste Regulations (e.g. BC's Hazardous Waste Regulation, O.Reg. 347 in Ontario); Montgomery & Runger, Applied Statistics and Probability for Engineers (for Q1–Q5's basic-statistics content).
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
Student's t is a probability distribution (and the associated test statistic) used in place of the normal (z) distribution when the population standard deviation is unknown and must itself be estimated from a small sample. The t-statistic, $t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$, has the same bell shape as the normal distribution but heavier tails, reflecting the additional uncertainty introduced by estimating σ from the sample itself rather than knowing it exactly; its shape depends on the degrees of freedom ($n-1$) and converges to the standard normal distribution as n grows large (in practice, by about $n \geq 30$). It is used to construct confidence intervals for a mean and to test hypotheses about a mean (or the difference between two means) whenever the sample size is small and σ is not independently known — the typical situation for a limited environmental-monitoring or industrial-waste sampling program.
A practical example: comparing an industrial effluent's mean BOD from a 6-sample monitoring event against a 300 mg/L permit limit uses a one-sample t-test rather than a z-test precisely because six samples is far too few to treat the sample standard deviation as if it were the true, known population σ. Using the z-distribution instead in this situation would understate the true sampling uncertainty and could lead to declaring a false compliance violation (or, just as problematically, a false pass) with more confidence than the small sample size actually supports. As the sample size grows into the dozens, the difference between the t and z critical values shrinks to the point of being practically negligible, which is why the distinction matters most precisely in the small-sample regime typical of a limited monitoring event.
