In general, we can never be 100% certain in our conclusions. However, we can estimate the likelihood that the same pattern of results will be seen again if we were to repeat the study several times. Since our sample statistics are only estimates of the overall population parameters, our best estimate of the parameter will result if we take a random sample or chunk of the overall population of individuals. That is, no individual should have a greater chance of being selected for the study than any other individual. If the proper procedure is followed, we can use our sample statistics to estimate the probability that the sample we selected is representative of the population in general.

The diagram to the right demonstrates the differences between a sample mean and a population mean. The closer the sample mean is to the population mean, the more representative the sample is of the total population. Therefore, we can be more confident that our results are typical of what would be found if we had tested the entire population.

Random assignment goes beyond a random sample in that it states that each individual that is selected for the study should have an equal chance of being placed into any of the treatment groups. This important because it allows you to infer that the treatment you are giving the participants is actually causing the differences that you find rather than some other factor that you are not controlling.

1. It allows us to gain some idea of the set of possible outcomes of an experiment, and
2. To understand the likelihood of any particular outcome.

All the possible outcomes of a sampling experiment is defined as the sample space (S). For example, the sample apce for tossing 3 coins has 8 possible outcomes: HHH, THH, HTH, HTT, THT, HHT, TTH, and TTT. For some situations, however, we can have an infinite number of outcomes (e.g. a sampling of 10 children's IQ scores). In such cases, we use some theoretical distribution as the sample space, such as a normal distribution.

A single experimental outcome is called an elementary event. In the above example of coin tossing, HHH was an elementary event. A particular set of elementary events is labeled an event class. For example, event class A for the coin tossing situation could be "all heads or all tails"--a class where there are two elementary events (HHH, TTT).

The probability of some event class occuring can be defined as a ratio of elementary events in the particular class to the total number of number of possible outcomes in the sample space. Thus, the probability of event class A in the above example is:

Note that probabilities can range from 0 to 1.0. If in your calculations you arrive at a probability greater than 1.0, you did something wrong.

The union of two sets refers to the combination of those members which are in either set or both sets. That is, the union combines all members of A and B into one set. For the above example with kings and jacks, the probability of this union can be found using the additive rule:

p(A or B) = p(A) + p(B) = 4/52 + 4/52 = 8/52

However, two sets often share a common member. For example, class A could be "all kings" while class B is "all hearts." In this case, one card--the king of hearts--is in both classes.

Thus, when we compute the union for this case, we must correct for this overlap:

p(A or B) = p(A) + p(B) - p(overlap) = 4/52 + 4/52 - 1/52 = 7/52

The overlap or shared member is known as the intersection between two classes. In some cases, such as above, this is relatively easy to calculate; more often, this is difficult because of a very large sample space and multiple factors of interest (e.g. being bald and being old). What needs to be considered further are the different types of probability, to which we turn to next.