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<br />4 <br /> <br />TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS <br /> <br />Statistical inference <br /> <br />We have 54 years of record on the Rappahan- <br />nock River of Vll'ginia and might ask two ques- <br />tions about the mean flow. First, what is the <br />mean flow for the period of record? This is a <br />unique value which can easily be computed. <br />The second question, what is the mean flow of <br />the stream?, cannot be answered definitely. We <br />can only assume that the mean of the 54-year <br />sample is an estimate of the true (population) <br />mean. In other words, we infer the population <br />characf-!'ristics from those of a sample from that <br />population, <br />Statistical inference is based on the theory <br />of sampling, From a population of known char- <br />acteristics many samples are drawn (either <br />actually or conceptually), and the relation of <br />the sample characteristics to the population <br />characteristics is defined, <br />Sampling theory requires use of the concept <br />of a probability distribution, Assume that the <br />distribution of some random variable is normal <br />with mean 1', and standard deviation, u, ss <br />shown in figure 4. (The term "random," as <br /> <br /> <br />p. <br /> <br />Figure 4.-Normal distribution. <br /> <br />used here, means that the probability of drawing <br />anyone item of the population is the same ss <br />for any other,) <br />Now suppose we take many samples of size <br />N from this distribution, compute the mean of <br />each of these samples, and compute the mean <br />and variance of these sample means. The dis- <br />tribution of the means of samples of size N is <br />superposed on the original distribution in figure <br />5. It can be shown that the distribution of <br />the means is centered at I' and that the stan- <br />dard deviation of the distribution of means is <br />u/.,fN, Therefore, the mean of the means of <br /> <br />. <br /> <br />Distribution of means <br />of size N <br /> <br /> <br /> <br />jJ. <br /> <br />Figure 5.-Distribution of means of samples from a norma. <br />distribution. <br /> <br />samples of size N is an unbissed estimate of 1'. <br />Furthermore, the mean of one sample is an <br />unbi.sed estimate of 1'. Consequently, we infer <br />that the sample mean, X, is an estimate of the <br />population mean. Obviously, if we used other <br />samples we would obtain different estimates <br />of the population mean, <br />From a single sample, we can appraise the <br />reliability of the estimate, X, of the population <br />mean, The distribution of means of values <br />drawn from a normal distribution is normal. <br />Consequently, two-thirds of the values should <br />fall within one standard deviation (u/M <br />on each side of the mean. However, we do not <br />know u so we have to substitute S for it (where <br />S is the standard deviation computed from the <br />sample). The distribution of X having a <br />standard deviation of S/..[N i6 known as the <br />Student's t distribution, values of which are <br />tabulated in statistics texts for various sizes <br />ofN, <br />Suppose now that we have K samples of <br />size N and have defined K different sampling <br />distributions of the mean of size N, For each <br />sampling distribution, we can define a mean <br />and a range of reliability, and we are interested <br />in whether such a range includes the true mean <br />1'. Considering the range ss a random interval, <br />we may state that the probability (P) that <br />the random interval includes I' is l-e, where e <br />is the level of significance. Mathematically, for <br />e==0.32, <br /> <br />. <br /> <br />P[(X -u/...jN) <I'<(X +u/"/N)]=I-e=O,68, <br /> <br />. <br />