Pearson Correlation

    The simplest way to determine whether there is an association between two variables is to plot them in a scatter plot and visually decide whether both vary together based on the shape and spreas of the plotted points.  The correlation coefficient, or Pearson coefficient, 

gives a numerical value to this method.  The correlation coefficient varies between -1 and 1.  The farther the value is form 1, the greater the association between the variables.  A negative value indicated negative association while a positive value indicates positive association.

    The t test for correlation can be used to test the hypothesis that the population correlation coefficient is actually zero.  The test statistic is 

The critical value for the test can be found in the t table with n-2 degrees of freedom and the desired alpha level.  If the test statistic is inside the critical value, the null hypothesis that the population correlation coefficient is zero is not rejected. 

    If the hypothesized population coefficient is not zero, the sample value is transformed by Fisher’s z transformation

so that the z distribution can be used to test the hypothesis.  The test statistic is

The calculated test statistic is compared to the critical value from the z table at the appropriate alpha level.  It the test statistic is closer to zero than the critical value, the null hypothesized that the population is some hypothesized value is not rejected.