A Pearson coefficient is used to test the linearity of a two-variable continuous distribution. What this means is that each of the variables you are comparing (in this case, IQ and GPA) are measured for every subject, and you are testing whether there is a predictive correlation between the two variables.
The value of the Pearson coefficient ('r') varies between -1 and 1. 0 is a perfectly non-linear relationship. 1 is a perfectly positive linear (all the samples lie on a perfect line with positive slope), and -1 is a perfectly negative linear correlation.
On to the questions:
- Evaluate the correlational result and identify the strength of the correlation.
A correlation coefficient of 0.75 means there is a strong positive correlation between IQ and GPA. This means that higher GPAs often correspond with higher IQs.
- Examine the assumptions and limitations of the possible connection between the researcher’s chosen variables.
The Pearson coefficient only tests for linear correlations between variables, not correlations of any other type (higher order polynomials, exponentials, etc.). This simply describes how linear the relationship is (how tightly the variance is distributed around a perfectly linear relationship), not the slope of the correlation. The slope of the correlation describes the magnitude with which changing one variable would correlate with a change in the other. Additionally, this test assumes each variable is normally distributed (Gaussian) and the samples contain no outliers.
In the case of your particular variables, GPA seems like it might not satisfy the conditions of normality since it is bounded, likely left tailed.
- Identify and describe other statistical tests that could be used to study this relationship.
The Spearman's correlation coefficient does not assume a linear relationship between two variables; rather, it "ranks" (assigns a relative position to) each variables independently and describes how likely a sample is to be the "Nth" rank in both variable A and variable B. Additionally, a Spearman's correlation can be used when one or both of the variables is non-normally distributed.