An Introduction to the theory of statistics

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Monograph

Identifikator:: 1751730271

URN:: urn:nbn:de:zbw-retromon-127610

Document type:: Monograph

Author:: Yule, George Udny http://d-nb.info/gnd/12910504X

Title:: An Introduction to the theory of statistics

Edition:: 8. ed. rev

Place of publication:: London

Publisher:: Griffin

Year of publication:: 1927

Scope:: XV, 422 S; Ill., Diagr

Digitisation:: 2021

Collection:: Economics Books

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Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: Part III. Theory of sampling

Collection:: Economics Books

> THEORY OF STATISTICS. dependent on the result of drawing the first. The disturbance can only be eliminated by drawing from a bag containing a number of balls that is infinitely large compared with the total number drawn, or by returning each ball to the bag before drawing the next. In this chapter our attention will be confined to the case of independent sampling, as in coin-tossing or dice- throwing—the simplest cases of an artificial kind suitable for theoretical study and experimental verification. For brevity, we may refer to such cases of sampling as simple sampling: the implied conditions are discussed more fully in § 8 below. 4. If we may regard an ideal coin as a uniform, homogeneous circular disc, there is nothing which can make it tend to fall more often on the one side than on the other; we may expect, there- fore, that in any long series of throws the coin will fall with either face uppermost an approximately equal number of times, or with, say, heads uppermost approximately half the times. Similarly, if we may regard the ideal die as a perfect homogeneous cube, it will tend, in any long series of throws, to fall with each of its six faces uppermost an approximately equal number of times, or with any given face uppermost one-sixth of the whole number of times. These results are sometimes expressed by saying that the chance of throwing heads (or tails) with a coin is 1/2, and the chance of throwing six (or any other face) with a die is 1/6. To avoid speaking of such particular instances as coins or dice, we shall in future, using terms which have become conventional, refer to an event the chance of success of which is p and the chance of failure ¢. Obviously p+¢=1. 5. Suppose we take IV samples with » events in each. What will be the values towards which the mean and standard-deviation of the number of successes in a sample will tend? The mean is given at once, for there are N.n events, of which approximately pNn will be successes, and the mean number of successes in a sample will therefore tend towards pn. As regards the standard- deviation, consider first the single event (n=1). The single event may give either no successes or one success, and will tend to give the former gk, the latter pXV, times in XN trials. Take this frequency-distribution and work out the standard-deviation of the number of successes for the single event, as in the case of an arithmetical example :— Frequency f. Successes &. J 2 g rN ' ] = y 256 AN

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