1. Research Methods for Counselors COUN 597 University of Saint Joseph Class # 9 Copyright © 2014 by R. Halstead. All rights reserved.

2. Class Objectives  Salkind Chapter 13 - Correlational Tests  Salkind Chapter 14 - Chi-square

3. Testing Correlation Coefficients  Let’s think way back in the semester to that time when we were learning about correlation.  You will recall that the correlation coefficient expressed the strength of relationship between two sets of distributed scores.  We can establish if the correlation between two variables is significant.  This is a very easy procedure very similar to the ones that we have seen in recent weeks.

4. Steps in Computing the Test Statistic  Step 1: A statement of the null hypothesis and research hypotheses – the is no relationship between the variables and the alternative.  The Null Hypothesis Ho:  xy = 0 The Research Hypothesis H 1 :  xy = 0  Note: A directional research hypothesis is reflected in the direction (+ or -) of the correlation  Step 2: Set the level of risk (or level of significance) upon which your decision will be made to reject the null hypothesis and risking Type I Error.

5. Steps in Computing the Test Statistic  Step 3: Select the appropriate test statistic using Salkind’s flow chart.  Step 4: Consult the appropriate table to determine the Critical Value needed for rejecting the null hypothesis for the statistic you are using.  To use the table you must determine the Degrees of Freedom (df). Here, df is determined by n-2.

6. Steps in Computing the Test Statistic  Step 6: Compare the Obtained Value and the Critical Values or correlation for rejecting the null hypothesis  Step 7: Commit to a decision. If the Obtained Value is more extreme than the critical value (further out in the tail of the normal curve) the test has been met. We can reject the null hypothesis and conclude that the correlation between the two variables is Significant and can risk saying the strength of the relationship is not do a chance occurrence.

7. r (28) =.393, p <.05  In the Salkind text we see the expression of the results. So what does that mean?  r represents the test statistic that was used  28 is the number of degrees of freedom .393 is the obtained value using the Person Product Moment formula introduced in Chapter 5  p <.05 indicates that the probability is less than 5% on any one test of the null hypothesis that the relationship between the two variables is due to chance alone.

8. Significance vs. Meaningfulness  Again we must ask, “Just because something is found to be significant, does that mean it is meaningful?  Let’s take a look again at this problem. Say we use the same correlation as in our example (.393). If we square it to get the Coefficient of Determination, we find that we can account for 15.4% of the variance. This means that 84.6% of the variance is left unaccounted for which is really worth noting when we are trying to determine meaningfulness.

9. Chi-Square  Probability Probaschmility! What about when we can’t use parametric tests?  Simple. We use nonparametric tests (also called distribution-free statistics).  Nonparametric statistics allows you to effectively deal with data that come in the form of frequencies or proportions.

10. One-Sample Chi-Square  Chi-square (“Ki – square) allows you to determine if what you observe in a distribution of frequencies would be what you would expect to occur by chance.  A one-sample Chi-Square includes only one dimension (logically, a two-sample Chi-square includes two dimensions). 2 2 (O-E) X = E

11. Steps in Computing Chi-Square  Step 1: A statement of the null hypothesis and research hypotheses - no difference in the frequency or the proportion of occurrences in each category.  The Null Hypothesis Ho: P 1 = P 2  The Research Hypothesis H 1 : P 1 = P 2  Step 2: Set the level of risk (or level of significance) upon which your decision will be made to reject the null hypothesis and risking Type I Error (e.g.,.05).

12. Steps in Computing Chi-Square  Step 3: Select the appropriate test statistic. Here we do not use Salkind’s statistic selection chart because we are working with nonparametric data.  Step 4: Consult the appropriate table to determine the Critical Value needed for rejecting the null hypothesis for the statistic you are using.  See Table B.5 Critical Values for the Chi-Square

13. Steps in Computing Chi-Square  Step 6: Compare the Obtained Value and the Critical Value.  Step 7: Commit to a decision. If the Obtained Value is more extreme than the Critical Value then the null hypothesis will be rejected thus concluding that the two proportions are different.

14. 2 X (2) = 20.6, p <.05  In the Salkind text we see the expression of the results.  Chi-Square is the test statistic that was used  2 is the number of degrees of freedom  20.6 is the obtained value  p <.05 indicates that the probability is less than 5% that on any one test of the null hypothesis that the frequency of occurrence (votes) is equally distributed across all categories.