Geschichte der volkswirtschaftlichen Lehrmeinungen

An Introduction to the theory of statistics

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Monograph

Identifikator:: 1010911953

URN:: urn:nbn:de:zbw-retromon-20641

Document type:: Monograph

Author:: Gide, Charles http://d-nb.info/gnd/117543985; Rist, Charles http://d-nb.info/gnd/172332966

Title:: Geschichte der volkswirtschaftlichen Lehrmeinungen

Edition:: Nach der zweiten durchgesehenen und verbesserten Ausgabe, herausgegeben von Franz Oppenheimer

Place of publication:: Jena

Publisher:: Verlag von Gustav Fischer

Year of publication:: 1913

Scope:: 1 Online-Ressource (XXII, 828 Seiten)

Digitisation:: 2018

Collection:: Economics Books

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Document type:: Monograph

Structure type:: Chapter

Title:: Erstes Buch. Die Begründer

Collection:: Economics Books

XIV.—REMOVING LIMITATIONS OF SIMPLE SAMPLING. 277 greater than 0-5, greater than is possible for positive errors. The assumption is not, however, likely as a rule to lead to a serious mistake ; as stated at the commencement of this paragraph, the point is of importance only when = is small, for when # is large the distribution tends to become sensibly symmetrical even for values of p differing considerably from 0-5. (CF. Chap. XV. for the properties of the limiting form of distribution.) 2. In the second place, the student should note that, where we were unable to assign any a priori value to p, we have assumed that it is sufficiently accurate to replace p in the formula for the standard error by the proportion actually observed, say Where 7 is large so that the standard error of 2» becomes small relatively to the product pg the assumption is justifiable, and no serious error is possible. If, however, n be small, the use of the observed value = may lead to an under- or over-estimation of the standard error which cannot be neglected. To get some rough idea of the possible importance of such effects, the approximate standard error ¢ may first be calculated as usual from the observed proportion 7, and then fresh values recalculated, replac- ing 7 by 7+3e. It should be remembered that the maximum value of the product pg is given by »=¢=05, and hence these values, if within the limits of fluctuations of sampling, will give one limiting value for the standard error. The procedure is by no means exact, but may serve to give a useful warning. Thus in Example iii. of Chap. XIII. the observed proportion of tall plants is 29/68, or, say, 43 per cent. The standard error of this proportion is 6 per cent., and a true proportion of 50 per cent. is therefore well within the limits of fluctuations of sampling. The maximum value of the standard error is therefore i (20x50) = 606 per cent. On the other hand, the standard error is unlikely to be lower than that based on a proportion of 43 — 18 =25 per cent., i (BX) =5'25 per cent. 3. The two difficulties mentioned in § 1 and 2 arise when n, the number of cases in the sample, is small. The interpretation of the value of the standard error is also more limited in this case than when = is large. Suppose a large number of observa- tions to be made, by means of samples of # observations each, on different masses of material, or in different universes, for each of which the true value of p is known. On these data we could

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