Statistics for Environmental Engineers

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A similar confidence interval can be defined for the true population mean:


y — tv.al2Sy < n < y + tv.al2Sy


If the standard n0 falls outside this interval, it is declared to be different from the true population mean n, as estimated by y, which is declared to be different from n0.

Case Study Solution


The concentration of the standard specimens that were analyzed by the participating laboratories was


1.2    mg/L. This value was known with such accuracy that it was considered to be the standard: n0=


1.2    mg/L. The average of the 14 measured DO concentrations is y = 1.4 mg/L, the standard deviation is s = 0.31 mg/L, and the standard error is sy = 0.083 mg/L. The difference between the known and measured average concentrations is 1.4 1.2 = 0.2 mg/L. A t-test can be used to assess whether 0.2 mg/L is so large as to be unlikely to occur through chance. This must be judged relative to the variation in the measured values.


The test t statistic is t0= (1.4 1.2)l0.083 = 2.35. This is compared with the t distribution with v = 13 degrees of freedom, which is shown in Figure 16.1a. The values t = 2.16 and t = +2.16 that cut off 5% of the area under the curve are shaded in Figure 16.1. Notice that the a = 5% is split between 2.5% on the upper tail plus 2.5% on the lower tail of the distribution. The test value of t0= 2.35, located by the arrow, falls outside this range and therefore is considered to be exceptionally large. We conclude that it is highly unlikely (less than 5% chance) that such a difference would occur by chance. The estimate of the true mean concentration, y = 1.4, is larger than the standard value, n0= 1.2, by an amount that cannot be attributed to random experimental error. There must be bias error to explain such a large difference.

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