Samples
and
populations

Probability
theory and statistical inferrence

Inferring a population
mean

Tests of
statistical significance

Tests of
differences between
means

Coefficient
of
correlation

Caution
with
association

Chi square

Other
tests of
significance

Caution in
using
statistical
test results

Aids

Tests of statistical significance

Analyses are frequently done to see if there is a relationship between two variables.    Almost always some relationship is found.   The question then is whether the observed result is large or strong enough to say that there is a relationship between the variables in the populations from which the samples were drawn.    In making this decision you have to choose between two explanations. One explanation is that the relationship is the result of chance variations in the selection of the sample and, therefore, the relationship found in the sample data does not describe a true relationship between the variables in the population. The other explanation is that the relationship is too strong to be explained by sampling error and, therefore, most likely reflects a genuine relationship between the variables in the population. A statistical test of significance is used in choosing between these two explanations. A test of significance is based on testing the null hypothesis.

Logic of the null hypothesis

Well-established procedures are used in making this choice. The first step is to state a null hypothesis which is the standard statistical way of saying that there is no relationship between two variables or that there are no differences between two measures of central tendency. Forms of the null hypothesis are:

There is no relationship between variable X and variable Y. For example, years of schooling for females is not related to their fertility.

There is no difference between the means or percentages in one sample as compared to another sample. For example, there is no difference in means of males and females for attitudes toward gender equality.

After a null hypothesis is formulated, it is subjected to a statistical test of significance. This process is called hypothesis testing. Later in this chapter we describe a few of the many tests of significance available to social researchers. On the basis of the statistical test of significance, the null hypothesis is either accepted or rejected. When a researcher decides that the difference or relationship found could be due to chance variations, the null hypothesis is accepted. In this case, the choice is that the independent variable, for example, had no effect on a dependent variable or that two means are not statistically different. When the result of a statistical test suggests that the difference or relationship is too large to have occurred due to chance, the null hypothesis is rejected. In that case, we conclude that the difference or relationship we observed could not have occurred due to chance. The finding is assumed to reflect a real relationship or a real difference between two variables in the population.

You may wonder whatever happened to the idea of a hypothesis established at the beginning of a study. When a researcher decides to use a statistical test of significance, the fate of the original hypothesis depends on the outcome of the test of the null hypothesis. The original hypothesis becomes an alternative hypothesis: that is, it is the alternative to be considered only if the null hypothesis is rejected.

The null and alternative hypotheses

The original research hypothesis, formed at the beginning of a study, states what the investigator expects to find. Usually this hypothesis expresses a positive or negative relationships between two variables or that a difference exists between two samples with respect to some variable. When the null hypothesis is accepted, the original or now called the alternative hypothesis is rejected. Acceptance of the null hypothesis indicates that there is not a sufficient basis for accepting the alternative hypothesis. When there is a strong enough basis for rejecting the null hypothesis, the alternative hypothesis established initially to guide the research, can be accepted.

All statistical tests of significance work on this reverse way of thinking. On the basis of probability theory, we can only give the odds or chances that a finding is not due to chance. We cannot say absolutely that a relationship between two variables exists; we can only give evidence that it is unlikely that the relationship was due to chance. Statistical tests of significance are used in deciding whether to accept or reject the null hypothesis. They allow us to express results in terms of levels of confidence.

To repeat, in statistical analysis, we can never prove anything. Instead, we offer data that cast doubt on the negative argument and, therefore, provide a sufficient basis for reaching the opposite conclusion — that there is a relationship between two variables. If you do not understand this point, you will not understand any of the rest of this chapter. This kind of thinking takes getting used to, but by the end of this chapter you should master it.