Statistics for Environmental Engineers

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0.0594


0.0582


0.0571


0.0559


1.6


0.0548


0.0537


0.0526


0.0516


0.0505


0.0495


0.0485


0.0475


0.0465


0.0455


1.7


0.0446


0.0436


0.0427


0.0418


0.0409


0.0401


0.0392


0.0384


0.0375


0.0367


1.8


0.0359


0.0351


0.0344


0.0366


0.0329


0.0322


0.0314


0.0307


0.0301


0.0294


1.9


0.0287


0.0281


0.0274


0.0268


0.0262


0.0256


0.0250


0.0244


0.0239


0.0233


2.0


0.0228


0.0222


0.0217


0.0212


0.0207


0.0202


0.0197


0.0192


0.0188


0.0183

It is convenient to work with standardized normal deviates, z = (y — n)/a, where z has the distribution N(0, 1), because the areas under the standardized normal curve are tabulated. This merely scales the data in terms of the standard deviation instead of the original units of measurement (e.g., concentration). A portion of this table is reproduced in Table 2.1. For example, the probability of a standardized normal deviate exceeding 1.57 is 0.0582, or 5.82%.

The t Distribution


Standardizing a normal random variable requires that both n and a are known. In practice, however, we cannot calculate z = (y — n)/a because a is unknown. Instead, we substitute s for a and calculate the t statistic:


t = y-n


s


The value of n may be known (e.g., because it is a primary standard) or it may be assumed when constructing a hypothesis that will be tested (e.g., the difference between two treatments is assumed to be zero). Under certain conditions, which are given below, t has a known distribution, called the Student’s t distribution, or simply the t distribution.

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