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When evaluating the relationship between quantitative variables using Pearson's $$r$$, the appropriate statistical procedure is a hypothesis test of the correlation coefficient. This test shares the fundamental logic of other null hypothesis tests but specifically assesses whether a true, non-zero correlation exists within the broader population.

Hypothesis Test of the Correlation Coefficient

In a hypothesis test of a correlation coefficient, the null hypothesis posits that there is absolutely no relationship between the variables in the overall population. This parameter is mathematically represented using the Greek letter rho ($$ho$$), where the null hypothesis is precisely $$ho = 0$$.

Null Hypothesis for a Correlation Test

Consider a study investigating the correlation between people's calorie estimates and their weight. Since there is no expectation about the direction of the relationship, a two-tailed test is conducted. For a sample of $$22$$ students, Pearson's $$r$$ is computed to be $$-.21$$. Using statistical software, the $$p$$-value is found to be $$.348$$. Because the $$p$$-value is greater than $$.05$$, the null hypothesis is retained, concluding that there is no statistically significant relationship between the variables. Alternatively, computing the test by hand involves finding the critical value. For $$22 - 2 = 20$$ degrees of freedom, the critical value is $$.444$$. Since the absolute value of the sample correlation coefficient ($$.21$$) is less extreme than the critical value ($$.444$$), this confirms that the $$p$$-value is greater than $$.05$$, leading to the same conclusion to retain the null hypothesis.

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