Testing a Population Correlation with a t-Test

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What are $\rho$ and $r$?

$\rho$ is the population correlation coefficient, usually unknown. $r$ is the sample correlation coefficient calculated from observed data and used to estimate $\rho$.

What hypotheses test whether population correlation is significant?

The two-sided test uses $H_0: \rho=0$ and $H_a: \rho\ne0$. The alternative represents a nonzero population linear correlation.

What is the correlation test statistic and its degrees of freedom?

$t_c=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$, with $n-2$ degrees of freedom.

How is the t-test decision made?

Compare $|t_c|$ with the positive two-tailed critical t-value for the chosen significance level and $n-2$ degrees of freedom. Exceeding it means reject $H_0$; otherwise, fail to reject $H_0$.

Does a significant correlation prove causation?

No. It provides evidence of a nonzero linear association, but does not establish that one variable causes the other. Confounding or other explanations may account for the association.

Quiz questions

1. What does the population correlation coefficient $\rho$ describe?Answer: The direction and strength of a linear relationship in the population.
2. Which null hypothesis is used in a two-sided test of population correlation?Answer: $H_0: \rho=0$
3. For a correlation significance test with sample size $n$, what are the t-distribution degrees of freedom?Answer: $n-2$
4. In a two-tailed correlation test, what decision follows when $|t_c|$ exceeds the critical t-value?Answer: Reject $H_0$ and conclude the population correlation differs from zero.
5. What does a statistically significant correlation establish about causation?Answer: It does not establish that either variable causes changes in the other.