Statistics for Environmental Engineers

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Statistical inference involves making an assessment from experimental data about an unknown population parameter (e.g., a mean or variance). Consider that the true mean is unknown (instead of being known as in Example 2.9) and we ask, “If a sample mean of 7.51 mg/L is estimated from measurements on 27 specimens, what is the likelihood that the true population mean is 8.00 mg/L?” Two methods for making such statistical inferences are to make a significance test and to examine the confidence interval of the population parameter.

The significance test typically takes the form of a hypothesis test. The hypothesis to be tested is often designated Ho. In this case, Ho is that the true value of the population mean is n = 8.0 mg/L. This is sometimes more formally written as Ho: n = 8.0. This is the null hypothesis. The alternate hypothesis is Ha: or n ^ 8.0, which could be either n < 8.0 or n > 8.0. A significance level, a, is selected at which the null hypothesis will be rejected. The significance level, a, represents the risk of falsely rejecting the null hypothesis.

The relevant t statistic is:

f = statistic — E( statistic)

JV (statistic)

where E(statistic) denotes the expected value of the statistic being estimated and V(statistic) denotes the variance of this statistic.

A t statistic with v degrees of freedom and significance level a is written as tva.

Example 2.10

Use the nitrate data to test the hypothesis that n = 8.0 at a = 0.05. The appropriate hypotheses are Ho: n = 8.0 and Ha: n < 8.0. This is a one-sided test because the alternate hypothesis involves n on the lower side of 8.0.

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