When utilizing the sample correlation coefficient to compute a $$t$$ score for null hypothesis testing, how are the degrees of freedom ($$df$$) determined?

When utilizing the sample correlation coefficient to compute a $$t$$ score for null hypothesis testing, the degrees of freedom ($$df$$) are determined by subtracting two from the total sample size ($$N$$). This calculation is expressed mathematically as $$df = N - 2$$.

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The test of the correlation coefficient is the appropriate null hypothesis test for evaluating statistical relationships between quantitative variables when Pearson's $$r$$ is used to describe the strength of those relationships. While it is possible to use the sample correlation coefficient to compute a $$t$$ score and proceed with a standard $$t$$-test, the correlation coefficient is mathematically structured so that it can also be treated directly as its own test statistic to find the $$p$$-value.

Test of the Correlation Coefficient

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KPU Research Methods in Psychology - 4th American Edition

In a test of the correlation coefficient, the null hypothesis posits that there is no relationship between the variables in the broader population. This is formalized using the Greek lowercase rho ($$\rho$$) to represent the population parameter: $$\rho = 0$$. Conversely, the alternative hypothesis asserts that a relationship does exist in the population: $$\rho \neq 0$$. Researchers can specify this as a two-tailed test if they do not expect a specific direction, or a one-tailed test if they theoretically predict the relationship will be strictly positive or strictly negative.

Null and Alternative Hypotheses for the Correlation Coefficient

Degrees of Freedom (Correlation Coefficient)

When conducting a test of the correlation coefficient by hand, researchers can use a statistical table to find the critical values of $$r$$ for various sample sizes at a predetermined alpha level, such as $$\alpha = .05$$. If the calculated sample correlation coefficient is more extreme than the critical value retrieved from the table, the result is statistically significant, meaning there is sufficient evidence to reject the null hypothesis and conclude a relationship exists in the population.

Critical Values of r

To illustrate a test of the correlation coefficient, consider a health psychologist investigating the correlation between people's calorie estimates and their weight. Having no directional expectation, she conducts a two-tailed test. For a sample of $$22$$ students, she computes Pearson's $$r$$ as $$-.21$$. Statistical software provides a $$p$$-value of $$.348$$. Because this $$p$$-value is greater than $$.05$$, she retains the null hypothesis, concluding there is no relationship between the variables. Alternatively, computing the test by hand, the degrees of freedom are $$20$$ ($$22 - 2 = 20$$), and a statistical table indicates the critical value is $$.444$$. Because the calculated correlation coefficient ($$-.21$$) is less extreme than this critical value, the $$p$$-value is confirmed to be greater than $$.05$$, leading to the same conclusion to retain the null hypothesis.

Example Test of a Correlation Coefficient

When testing a correlation coefficient, a p-value greater than the alpha threshold indicates that the researcher should retain the null hypothesis. For example, if software reports a p-value of $$.348$$ and the threshold is $$.05$$, the result is not statistically significant and the researcher should conclude that the sample does not provide enough evidence for a population relationship.

Evaluating p-Values in a Correlation Test

When evaluating statistical relationships between quantitative variables, how can the correlation coefficient (Pearson's r) be utilized to determine the p-value in a null hypothesis test?

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