Testing Hypotheses: Review Prior to 2/3 Examination

REVIEW OF HYPOTHESIS TESTING Two types of hypotheses Research hypothesis: a succinct statement of a relationship that exists between two clearly specified variables. Nondirectional hypothesis: (Example: A person's activity within the community is related to his educational background.) Directional hypothesis: (Example: The greater the education, the greater the community activity.) Research hypotheses are supposed to be forcefully stated to permit their falsification or disproof, if possible. Social scientists are interested in the construction of theory to explain human behavior on the basis of causal laws.  But because correlation does not mean causation, the observance of correlations between variables as hypothesized never proves a theory.  Rather, theories becomes accepted as true as they consistently resist attempts to disprove them.  The lack of a hypothesized correlation means that causation does not exist, at least under the conditions of the theory as specified.  Thus, science advances really through attempts at disproving hypotheses rather than proving them, which requires that hypotheses be stated clearly in an attempt to facilitate attempts at disproof.   Null hypothesis: To facilitate the disproof of a research hypothesis, researchers adopt the ploy of constructing a null hypothesis, which is customarily a statement that no relationship exists between the variables that are linked together in the research hypothesis. A null hypothesis is usually constructed as the hypothesis to be tested by the data because the researcher usually cannot be specific enough about the strength of the relationship expected by the research hypothesis. The rejection of the null hypothesis of no relationship, therefore, would lend support of the research hypothesis by implication. Steps in testing hypotheses Making assumptions about the population from which the sample was drawn about the sampling procedure used.  Obtaining the sampling distribution of the statistics Many statistics, like the mean, have known sampling distributions. Knowing the sampling distribution, one can calculate the standard error of the statistic for samples of varying sizes.  Knowing the standard error means knowing the likely margin of error that one might encounter purely due to sampling error if the null hypothesis of no relationship were in fact true.  Considerations of the sampling distributions of the statistic are incorporated into formulas given to test the significance of a statistic. Selecting a significance level and critical region According to probability theory one might observe very rare events by the operation of purely chance factors -- e.g., we might observe high correlations between two variables for cases selected at random from some population when in fact there was no correlation between the variables in the population from which the cases were drawn.  Therefore, we must specify in advance some probability level such that the observance of so rare an event -- so deviant a correlation from the expected distribution of correlations given a true absence of a relationship between the variables -- would be interpreted as evidence that the assumption of no relationship (the null hypothesis) must be rejected.  This specified probability level is known as the level of significance and is referred to as the alpha value. It is commonly set to .05 or .01 in social research, which is largely a matter of convention.  A correlation coefficient or any observed test statistic that is "significant" at the .05 or .01 levels is one that is judged to be unlikely to occur simply on the basis of chance or sampling variation if there were no relationship within the population (or whatever the null hypothesis might assert). The value of alpha (e.g., .05 or .01), which expresses the level of significance, merely states how unlikely an event must be (i.e., how deviant a correlation must be) before observing the event will result in rejecting the null hypothesis.  This selected alpha value is associated with a critical value which marks the region of rejection) in the sampling distribution; if the observed test statistic falls within the region of rejection, the null hypothesis is rejected and it is presumed that there is some relationship between the variables. When one rejects the null hypothesis on the basis of a sample value, one is running the risk of rejecting the null hypothesis when it is true -- which means claiming that there really is a relationship between the variables when there is not.  This type of error is called a Type I error, and it is equal to alpha.  In general, reducing the size of the alpha value reduces the probability of making a Type I error; but as one reduces alpha, one runs the risk of making a Type II error.  A Type II error means accepting the null hypothesis when it is false, which in turn means claiming that there is no relationship between the variables (as specified by your null hypothesis) when in fact there is a relationship. The probability of making a Type II error is equal to beta.  In general, the probabilities of making alpha and beta errors are inversely related; minimizing the risk of a Type I error, increases the risk of a Type II error.  But the probability of a Type II error is not just the complement of a Type I error, and calculating the value of beta varies with the statistic, the sample size, and the hypothesis. In general, however, the beta error is related to sample size; the larger the sample, the less the risk of a Type II error.  Computing the test statistic: enter the table with the appropriate distribution Test Statistic Distribution z-score normal t-value t F-value F   Making the decision Compare the value of the test statistic with that expected under the null hypothesis, usually 0. Allow for the standard error of the statistic by applying the correct test. Enter the appropriate table to determine the "deviancy" of the observed statistic when tested against the null hypothesis and observe if the value falls within the critical region, allowing for one-tailed or two-tailed tests.