4. The Method of  3. Probability Distribution Functions  3.5 Student's t distribution

3.6 Confidence intervals

The distributions of the preceding sections are often used for inference. If the statistical character of the observations requires a particular distribution function, that function can be used to estimate the probability that the true mean is greater than some limit $\mu_2$ if the observed mean is $\overline{x}$.

First, consider the direct probability question: If the true mean is $\mu$ and the true standard deviation is $\sigma$, what is the probability that an observation will lie between x₁ and x₂? The answer is

$$P(x_1\le x\le x_2) = \int_{x_1}^{x_2} \phi(x) dx .$$ (3.17)

 
Often, the integrals of probability functions do not have analytical forms. Tables of cumulative values for the Gaussian distribution function and other standard forms are available in statistical handbooks and other reference books (e.g., Abramowitz and Stegun 1972), and values are also available on may computer systems through mathematical libraries. For example, Figure 2.1 showed the probability of exceeding various deviations for the Gaussian and Student-t distribution functions.

The inverse problem follows a similar procedure. If the experimentally determined estimate of the mean is $\overline{x}$and the estimate of the standard deviation is s, the procedure of the preceding paragraphs can be used to calculate the probability that a set of observations will give a mean $\overline{x}$ when the true mean is $\mu$. If the value of $\mu$is determined for which the observed mean $\overline{x}$ would be a deviation exceeded only with probability f, it is sometimes said that there is only a probability f that the true mean exceeds the limit $\mu$. The weaknesses in this argument is that the true standard deviation $\sigma$ is also unknown, and the estimate of probability depends on knowledge of this true standard deviation. If the experimental estimate s is used in place of $\sigma$, this is not a true inverse procedure and can lead to erroneous indications of confidence limits.

  SOURCES AND FURTHER READING  

Abramowitz, M. and I. A. Stegun, 1972: Handbook of Mathematical Functions. Dover Publications, New York, 1046 pp.

Bevington, P. R., 1969: Data Reduction and Error Analysis for the Physical Sciences. McGraw-Hill, New York, 336 pp.

Box, G. E. P., W. G. Hunter, and J. S. Hunter, 1978: Statistics for Experimenters. John Wiley and Sons, New York, 653 pp.

Brownlee, K. A., 1965: Statistical Theory and Methodology in Science and Engineering. John Wiley and Sons, New York, 590 pp.

Feller, William, 1950: An Introduction to Probability Theory and its Applications. John Wiley and Sons, New York, 461 pp. Statistics to Meteorology. Pennsylvania State University, 224 pp.

Press, W. H., Brian P. Flannery, S. A. Teukolsky, and W. T. Vetterling, 1992: Numerical Recipies in C. Second Edition, Cambridge University Press, Cambridge, 735 pp.

   
 4. The Method of  3. Probability Distribution Functions  3.5 Student's t distribution