Example 1.
The effect of four cooking temperatures on the fluffiness of omelets prepared from a mix was studied by randomly assigning five packages of mix to each of four cooking temperatures. This is an experiment because the independent variable of interest (cooking temperature) was controlled by the researcher. The experimental design was a completely randomized design, since the 20 packages of mix were randomly distributed among the cooking temperatures; this (together with some care in performing each of the 20 ``runs'') guarantees that the errors in the ANOVA/regression model are random and independent; it may not guarantee that they are normal, or have the same variability in each of the four cells (cooking temperatures) of the design.In a real industrial setting (say, to decide what instructions to write on boxes of omelet mix), probably all four cooking temperatures produce somewhat fluffy omelets. What is of interest is which one is better, not what the overall average fluffiness is. Since this is a common situation in ANOVA applications, ANOVA software often doesn't bother to report the grand mean, even in the ``grand mean plus effects'' parameterization.
If the packages were also chosen randomly from the company's output, it may be reasonable to assume that these results will be generalizable to the hypothetical population of all boxes of omelet mix of this type produced by this company.
Example 2.
A study of the effects of education and type of past employment experience of salespeople on their sales volumes was made by selecting a random sample of salespeople currently employed at the company and obtaining information on highest degree obtained, type of experience, and sales volume for each selected employee. This was an observational study because the independent variables (education, type of experience) was not controlled by the experimenter. Random sampling does guarantee that the errors will be random and independent (but again normality and equal variances across cells is harder).Because of the random sampling the results of this survey may generalize to a hypothetical population of employees like those working in sales for this company right now. However, it seems less likely that this is a population of interest---and even if it is, that it will remain stable over time so that the results can be used with future employees---than in Example 1.
Example 3.
An appliance manufacturer operates three training centers in the United States for training mechanics to service the company's products. At each center, two different training programs were studied, with trainees randomly assigned to each program. The outcome is a score on a standardized repair exam.
