By William Feller

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**Extra info for An Introduction to Probability Theory and its Applications**

**Sample text**

N Yn1 Yn2 Tn Here, Y is the observed score for a subject and T is the true score. Now assume that for a particular subject the scores are normally distributed about the true score, and that the true scores are also normally distributed. This leads to the following model for pairs of observations on a subject: The quantity is called the relability in the social sciences. In terms of the concepts and definitions from the previous section, the correlation coefficient ρis equal to the reliability.

A good rule of thumb is to assume that overlaps of 25% or less still suggest statistical significance. 6 SAMPLE SIZE CALCULATION FOR THE POISSON DISTRIBUTION Introduction The Poisson distribution is known as the law of small numbers, meaning that it deals with rare events. The term "rare" is undefined and needs to be considered in context. A rather elegant result for sample size calculations can be derived in the case of Poisson variables. It is based on the square root transformation of Poisson random variables.

80 that it will be picked up, if it occurs. Assume also that the test is two-sided—there could be a decrease in the number of deaths. Suppose the 35 average number of deaths per week in the United States (θ*) is 50,000—a reasonable number. The sample size is n = 1. 7) produces For this situation an additional number of deaths equal to would be needed to be reasonably certain that the assertion had been validated. All these calculations assume that the weeks were pre-specified without looking at the data.

### An Introduction to Probability Theory and its Applications by William Feller

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