This is just an Excerpt from a larger document, click here to view the entire document.Statistical Assumptions and Their Implications
Every statistical model has its own assumptions that have to be verified and met, to provide valid results. For example, deriving the small sample Student-t CI for the mean life of a device requires two assumptions: that the device lives are independent and Normally distributed. These two statistical assumptions must be met (and verified) in order for such CI to cover the true mean with the prescribed probability. If, as in data in Table 2, the data does not follow the assumed distribution, the CI is invalid and its coverage (confidence) may be different from the one prescribed. Fortunately, distribution model assumptions are associated with very practical and useful implications - the Normal and Lognormal distributions are no exceptions.
In practice, the assumption that the distribution of the lives of a device is Normal means that there are many, independent factors that are contributing to the final result. An analogy is the intelligence quotient (IQ). A human is the product of his or her socioeconomic level, upbringing, schooling, nutrition, inherited genes, health, etc. All these factors contribute to human intelligence. For this reason, IQ is usually Normally distributed, as are also (and for the same reason) height, weight, etc.
In addition, the Normal distribution has several specific characteristics. It is continuous, symmetric (mean = median = mode) and standarizable (by subtracting the mean and dividing by the standard deviation, we always obtain a unique Normal with mean zero and variance unit). Finally, the ranges defined by one, two, and three standard deviations above and below the mean cover 68%, 95%, and 99% of the population, respectively.
In what follows, we will use these statistical Normal distribution properties and its implications, to check and empirically validate the Normal assumptions of our data.