Introductory Statistics: Concepts, Models, and Applications
David W. Stockburger
Hypothesis tests are procedures for making rational decisions about the reality of effects.
Most decisions require that an individual select a single alternative from a number of possible alternatives. The decision is made without knowing whether or not it is correct; that is, it is based on incomplete information. For example, a person either takes or does not take an umbrella to school based upon both the weather report and observation of outside conditions. If it is not currently raining, this decision must be made with incomplete information.
A rational decision is characterized by the use of a procedure which insures the likelihood or probability that success is incorporated into the decision-making process. The procedure must be stated in such a fashion that another individual, using the same information, would make the same decision.
One is reminded of a STAR TREK episode. Captain Kirk, for one reason or another, is stranded on a planet without his communicator and is unable to get back to the Enterprise. Spock has assumed command and is being attacked by Klingons (who else). Spock asks for and receives information about the location of the enemy, but is unable to act because he does not have complete information. Captain Kirk arrives at the last moment and saves the day because he can act on incomplete information.
This story goes against the concept of rational man. Spock, being the ultimate rational man, would not be immobilized by indecision. Instead, he would have selected the alternative which realized the greatest expected benefit given the information available. If complete information were required to make decisions, few decisions would be made by rational men and women. This is obviously not the case. The script writer misunderstood Spock and rational man.
When a change in one thing is associated with a change in another, we have an effect. The changes may be either quantitative or qualitative, with the hypothesis testing procedure selected based upon the type of change observed. For example, if changes in salt intake in a diet are associated with activity level in children, we say an effect occurred. In another case, if the distribution of political party preference (Republicans, Democrats, or Independents) differs for sex (Male or Female), then an effect is present. Much of the behavioral science is directed toward discovering and understanding effects.
The effects discussed in the remainder of this text appear as various statistics including: differences between means, contingency tables, and correlation coefficients.
All hypothesis tests conform to similar principles and proceed with the same sequence of events.
Hypothesis testing is equivalent to the geometrical concept of hypothesis negation. That is, if one wishes to prove that A (the hypothesis) is true, one first assumes that it isn't true. If it is shown that this assumption is logically impossible, then the original hypothesis is proven. In the case of hypothesis testing the hypothesis may never be proven; rather, it is decided that the model of no effects is unlikely enough that the opposite hypothesis, that of real effects, must be true.
An analogous situation exists with respect to hypothesis testing in statistics. In hypothesis testing one wishes to show real effects of an experiment. By showing that the experimental results were unlikely, given that there were no effects, one may decide that the effects are, in fact, real. The hypothesis that there were no effects is called the NULL HYPOTHESIS. The symbol H0 is used to abbreviate the Null Hypothesis in statistics. Note that, unlike geometry, we cannot prove the effects are real, rather we may decide the effects are real.
For example, suppose the following probability model (distribution) described the state of the world. In this case the decision would be that there were no effects; the null hypothesis is true.
Event A might be considered fairly likely, given the above model was correct. As a result the model would be retained, along with the NULL HYPOTHESIS. Event B on the other hand is unlikely, given the model. Here the model would be rejected, along with the NULL HYPOTHESIS.
The SAMPLING DISTRIBUTION is a distribution of a sample statistic. It is used as a model of what would happen if
1.) the null hypothesis were true (there really were no effects), and
2.) the experiment was repeated an infinite number of times.
Because of its importance in hypothesis testing, the sampling distribution will be discussed in a separate chapter.
Probability is a theory of uncertainty. It is a necessary concept because the world according to the scientist is unknowable in its entirety. However, prediction and decisions are obviously possible. As such, probability theory is a rational means of dealing with an uncertain world.
Probabilities are numbers associated with events that range from zero to one (0-1). A probability of zero means that the event is impossible. For example, if I were to flip a coin, the probability of a leg is zero, due to the fact that a coin may have a head or tail, but not a leg. Given a probability of one, however, the event is certain. For example, if I flip a coin the probability of heads, tails, or an edge is one, because the coin must take one of these possibilities.
In real life, most events have probabilities between these two extremes. For instance, the probability of rain tonight is .40; tomorrow night the probability is .10. Thus it can be said that rain is more likely tonight than tomorrow.
The meaning of the term probability depends upon one's philosophical orientation. In the CLASSICAL approach, probabilities refer to the relative frequency of an event, given the experiment was repeated an infinite number of times. For example, the .40 probability of rain tonight means that if the exact conditions of this evening were repeated an infinite number of times, it would rain 40% of the time.
In the Subjective approach, however, the term probability refers to a "degree of belief." That is, the individual assigning the number .40 to the probability of rain tonight believes that, on a scale from 0 to 1, the likelihood of rain is .40. This leads to a branch of statistics called "BAYESIAN STATISTICS." While many statisticians take this approach, it is not usually taught at the introductory level. At this point in time all the introductory student needs to know is that a person calling themselves a "Bayesian Statistician" is not ignorant of statistics. Most likely, he or she is simply involved in the theory of statistics.
No matter what theoretical position is taken, all probabilities must conform to certain rules. Some of the rules are concerned with how probabilities combine with one another to form new probabilities. For example, when events are independent, that is, one doesn't effect the other, the probabilities may be multiplied together to find the probability of the joint event. The probability of rain today AND the probability of getting a head when flipping a coin is the product of the two individual probabilities.
A deck of cards illustrates other principles of probability theory. In bridge, poker, rummy, etc., the probability of a heart can be found by dividing thirteen, the number of hearts, by fifty-two, the number of cards, assuming each card is equally likely to be drawn. The probability of a queen is four (the number of queens) divided by the number of cards. The probability of a queen OR a heart is sixteen divided by fifty-two. This figure is computed by adding the probability of hearts to the probability of a queen, and then subtracting the probability of a queen AND a heart which equals 1/52.
An introductory mathematical probability and statistics course usually begins with the principles of probability and proceeds to the applications of these principles. One problem a student might encounter concerns unsorted socks in a sock drawer. Suppose one has twenty-five pairs of unsorted socks in a sock drawer. What is the probability of drawing out two socks at random and getting a pair? What is the probability of getting a match to one of the first two when drawing out a third sock? How many socks on the average would need to be drawn before one could expect to find a pair? This problem is rather difficult and will not be solved here, but is used to illustrate the type of problem found in mathematical statistics.