Frequency interpretation

The frequency interpretation of the interval estimates on the unknown amounts is given by the following ( 27 ) With at least 1- a confidence, based on the sampling characteristics of the observations on the standards, at least P proportion of the interval estimates made from a particular calibration will contain the true amounts. The Bonferroni inequality insures the 1-a confidence since the confidence interval about the regression line and the upper bound on cr are each performed using a 1- a/2 confidence coefficient. Hence, the frequency interpretation states that at least (1-a) proportion of the standard calibrations are such that at least P proportion of the intervals produced by the method cover the true unknown amounts. For the remaining a proportion of standard calibrations the proportion of intervals which cover the true unknown values may be less than P. 

Suppose that tlie valve slicks twice as oilen in tlie open position as it does in tlie closed position. Under tlie tlieoretical relative frequency interpretation, tlie probability assigned to clement 0 in S would be f, twice the probability... 

Depending on the problem, this would be done either once for a sufficiently long period of time or many times, so that an ensemble of solutions is obtained, from which the statistical properties can be calculated in accordance with a frequency interpretation of the stochastic process. 

Bellamy (110) and Colthup (111) are the best recognized and most extensive references on group frequency interpretation, and an easy-to-use introduction to group frequencies and a summary of major compound classes are in the Coblentz Society s "Desk Book" (112). 

We may give this result a frequency interpretation. Let U z) dz be the number of cells, per unit volume of cultme, having ages between z and z + dz. Then... 

We now adopt the frequency interpretation of probability. The number of cells of group x — dx at time t which become cells of group x at time t + dt is... 

The confidence interval has the following frequency interpretation. Suppose we obtain a 95% confidence interval from an experiment and statistical techniques. If we performed this experiment many times, then at least 95% of the time the interval will contain the true value of the parameter. This discussion of the confidence interval sounds similar to the previous discussion about estimators they are random variables, and they have a certain probability of being within certain intervals. 

INTRODUCTION TO GROUP FREQUENCIES INTERPRETATION OF INFRARED SPECTRA... 

Introduction to Group Frequencies Interpretation of Infrared Spectra 541... 

Many of these vibrational frequencies are associated with small groups of atoms that are essentially uncoupled from the rest of the molecule. The absorption bands that result from these modes, therefore, are characteristic of the small group of atoms regardless of the composition of other parts of the molecule. These vibrations are known as the group frequencies. Interpretation of infrared spectra of complex molecules based on group frequency assignments is an extremely powerful aid in the elucidation of molecular structure. 

Historically, statistical mechanics has been framed in terms of the frequency interpretation. JW Gibbs (1839-1903), American mathematical physicist and a founder of chemical thermodynamics and statistical mechanics, framed statistical mechanics as a counting problem. He envisioned an imaginary collection of all possible outcomes, called an ensemble, which was countable and could be taken to the limit of a large number of imaginary repetitions. To be specific, for die rolls, if you want to know the probability of each of the six outcomes on a roll, the ensemble approach would be to imagine that you had rolled the die N times. You would then compute the number of sequences that you expect to observe. This number will depend on N. On the other hand, if the outcomes can instead be described in terms of probabilities, the quantity N never appears. For example, the grid of cells described in Table 6.2 describes probabilities that bear no trace of the information that N = 1000 die roUs w-ere used to obtain Table 6.1. 

Bimolecular association reactions involving simple species generally have Arrhenius 4-factors much less than the collision frequency. Interpret this in terms of the discussion in Section 5.5. [For typical 4-factors see S. W. Benson, The Foundations of Chemical Kinetics (New York McGraw-Hill, 1960), p. 302.]... 

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