# Download e-book for kindle: Introduction to statistical inference by Keeping E.S. By Keeping E.S.

ISBN-10: 0486685020

ISBN-13: 9780486685021

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Example text

Consider a point Q with coordinates p (the true probability) and h (the empirical frequency) in the (h, p)-plane. Formula (2) represents the interior and the boundary of an ellipse which is tangent to the horizontal sides of the unit square at the end points of the diagonal (see Fig. 5). The width of the ellipse depends on g and n: the larger the number of trials the more narrow the ellipse. The location of Q depends on chance because the h coordinate depends on chance. The probability that Q lies inside or on the ellipse is approximately 1- 20( for each value of p.

The same holds for the difference x - y, except that the expectation is a-b. By repeated application, it follows that a sum of more than two normally distributed independent random variables is also normally distributed. Example 5. Corifidence limits for the expectation ol a normal distribution. First suppose we have made one observation XI of a normally distributed variable x. Suppose that the standard deviation (1 of the distribution is known, but the expectation a unknown. Can we find limits for the deviation XI - a such that the probability for XI - a to lie outside the limits is small?

The sum of 4 F. Wegmann, Die operative Behandlung der Bronchektasien, Diss. 39. 5 A curve which illustrates the shape of the error probability as a function of p in the limiting case of a rare event is shown in my note "Vertrauensgrenzen fUr unbekannte Wahrscheinlichkeiten", Sitzungsber. siichs. Akad. Wiss. 213. 6 C. J. Clopper and E. S. 404. 36 all the II. Probabilities and Frequencies ~ is h Lo l1\(P)= 1. (8) The one-sided confidence bound with error probability less than or equal to (J. is determined in the following manner.