Set Of Boundaries Research Articles

An Empirical Model Study of Cones of Depression Produced by Pumping Wellsa

AbstractIt is of the utmost importance that ground‐water users understand the basic concept that all the ground water within a single set of hydrologic and geologic boundaries is part of a single hydrologic system. It is therefore worthwhile for all ground‐water users to understand the manner in which ground water moves through the pore spaces of saturated rock and to understand the various configurations of the water table brought about by the pumping of water table wells. The movement of ground water is illustrated in two hydraulic models which were constructed in order to bring ground‐water flow into surroundings where it can be visually observed. The models consist of watertight plexi‐glass cases containing a porous consolidated mixture of sand and epoxy resin, which simulates a true sandstone. The models are used to study cones of depression produced by pumping wells. Photographic histories of several ground‐water flow conditions are illustrated.

Truncated sequential hypothesis tests

Through a careful examination of the equations by which Wald determines the values of the boundaries for tests of sequential hypotheses, we are able to obtain interesting relationships for the conditional probability distributions of the stage at which the test terminates. These enable us to study sequential tests in which the boundaries are functions of the sample number. We are particularly concerned with tests with convergent boundaries, and investigate a set of boundaries which approach those of a truncated Wald test. Approximate expressions for the expected sample number and probabilities of error are obtained for the tests considered. The obtained approximations apply best for gently tapering slopes. Extensions of our method can be applied to evaluate various ad hoc schemes for truncating tests and to the theory of tests with a varying parameter.

The Number and Width of Classes in the Chi-Square Test

Abstract The Helmert transformation of a set of observations {xj }, j = 1, 2, …, N is applied to give a direct approach to the problem of boundary determination in the case of the χ2-test for location of the normal distribution, i.e. the choice of that set of boundaries which maximizes the power of the test. It is found that no significant increase in power can be achieved by taking a number of classes greater than 20; and even a number as small as 12 is sufficient. Moreover, a power of one-half is achieved for a number of classes much smaller than that required by the Mann-Wald approach. For a preassigned number of classes, symmetrically situated about the origin, it is found that the optimum partition corresponds to equal class-width of about 0.4 standard deviation with pooling of the terminal classes, and this is slightly more powerful than the equal class-probability partition.