41

c h a p t e r 5

Statistical Inference

Now that we have collected our data, estimated some basic descriptive statistics and assumed a probability model for the underlying population or process, we are prepared to perform some statistical analysis that will allow us to compare populations and test hypotheses. For this text, we are only going to discuss statistical analysis for which the normal distribution is a good probability model for the underlying population(s). If the data or samples suggest that the underlying population or process is not well modeled by a normal distribution, then we will need to resort to other types of statistical analysis, such a nonparametric techniques [6], which do not assume an underlying probability model. Some of these tests include the Wilcoxon rank sum test, Mann–Whitney U test, Kruskal–Wallis test, and the runs test [6, 12]. More advanced texts provide details for administering these nonparametric tests. Keep in mind that if the statistical analysis presented in this text is used for populations or processes that are not normally distributed, the results may be of little value, and the investigator may miss important findings from the data.

Assuming our data represents a normally distributed population or process, we are now prepared to perform a variety of statistical tests that allow us to test hypotheses about the equality of means and variances across two or more populations. Remember that for normal distributions, only the mean and standard deviation are required to completely characterize the probabilistic nature of the population or process. Thus, if we are to compare two normally distributed populations, we need only compare the means and variances of the populations. If the populations or processes are not normally distributed, there may be other parameters, such as skew and kurtosis, which differentiate two or more populations or processes.

5.1COMPARISON OF POPULATION MEANS

One of the most fundamental questions asked by scientists or engineers performing experiments is whether two populations, methods, or treatments are really different in central tendency. More specifically, are the two population means, reflected in the sample means collected under two different experimental conditions, significantly different? Or, is the observed difference between two means simply because of chance alone?