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.

A Pearson's correlation of 0.75 can be regarded as a strong and positive association between our variables of interest, intelligence quotient (IQ) and grade point average (GPA). From this value, we can conclude that if one of these variables increases or decreases, the other variable is highly likely to show a corresponding increase or decrease.

Because the Pearson's correlation is a parametric statistic, its correct application requires that several assumptions are met. The data should be continuous, collected in related pairs, and free of outliers. The variables should be normally distributed, have a linear relationship, and have equal variances. A challenge with correlational studies is that they are descriptive in nature and merely suggest the presence of a relationship between the variables. From this data, we cannot establish a cause and effect relationship between IQ and GPA. It may be that the relationship between the two is explained by a third, unexplored variable.

In case the assumptions required for Pearson's correlation are unmet, Spearman's correlation, such as Spearman's rank correlation coefficient and Kendall's rank correlation coefficient, can be used. These are non-parametric counterparts of Pearson and do not require any distributional assumptions.

Correlations, measured by the Pearson's correlation coefficient (denoted by "r"), measure the strength and direction of the correlation of two variables. R is always between -1 (meaning a strong negative correlation) and 1 (meaning a strong positive correlation). If r=0, there is no correlation between the two variables. In this case, r=.75, which shows a strong positive correlation between IQ and grade point average (GPA). That is, as IQ increases, so does the GPA.

There are many assumptions that have to be satisfied in order to use Pearson's correlation coefficient. One of these is that the data have no significant outliers and that they are normally distributed (along the bell curve). However, some studies have shown that IQ is not not normally distributed (see Cyril Burt's "Is Intelligence Distributed Normally" from 1963). This may be in part because of poverty and other factors that result in IQ not being normally distributed in the United States.

Another test that can be used to measure the degree of correlation between two variables is Spearman's rank correlation. This test does not assume the normal distribution of data.

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