Statistical Inference (part II): The Normal and Related Distributions

The normal (or gaussian) is the most important probability distribution in the statistical inference process. By transforming the normal distribution to a standard distribution (z-distribution) it is possible to determine the probabilities where observations or values fall in certain intervals for variables with different means and standard deviations. Areas under the normal distribution may be represented in a table where the z is the number of standard deviations away from the mean. For example, the area under the normal distribution delimited by a z value of 1.96 to the left and 1.96 to the right of the mean corresponds to 95% of the total area of the normal distribution. Sampling distribution of estimates of population parameters may also be described by the normal distribution. It is important to note, however, that the estimation of a population mean based on the normal distribution is conditional to the assumption that the population standard deviation is a known parameter. The t distribution is used to infer about a population mean when the population standard deviation is estimated by the sample data. In statistical inference about proportions, the normal approximation of the binomial distribution may be used provided the data fit certain assumptions. Statistical methods that do not depend on the form of the distribution (distribution-free or nonparametric methods) and those based on the actual probability distribution, called exact methods, are often used in situations where the data do not comply with the assumption that the distribution of the estimate is approximately normal.