Statistics for Environmental Engineers

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A common and fundamental problem is making inferences about mean values. This chapter is about problems where there is only one mean and it is to be compared with a known value. The following chapters are about comparing two or more means.

Often we want to compare the mean of experimental data with a known value. There are four such situations:

1.    In laboratory quality control checks, the analyst measures the concentration of test specimens that have been prepared or calibrated so precisely that any error in the quantity is negligible. The specimens are tested according to a prescribed analytical method and a comparison is made to determine whether the measured values and the known concentration of the standard specimens are in agreement.

2.    The desired quality of a product is known, by specification or requirement, and measurements on the process are made at intervals to see if the specification is accomplished.

3.    A vendor claims to provide material of a certain quality and the buyer makes measurements to see whether the claim is met.

4.    A decision must be made regarding compliance or noncompliance with a regulatory standard at a hazardous waste site (ASTM, 1998).

In these situations there is a single known or specified numerical value that we set as a standard against which to judge the average of the measured values. Testing the magnitude of the difference between the measured value and the standard must make allowance for random measurement error. The statistical method can be to (1) calculate a confidence interval and see whether the known (standard) value falls within the interval, or (2) formulate and test a hypothesis. The objective is to decide whether we can confidently declare the difference to be positive or negative, or whether the difference is so small that we are uncertain about the direction of the difference.

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