Testing
Correlation Coefficients
Since the value of \(r\) is computed from data obtained from samples, there are two possibilities when r is not equal to zero: either the value of r is high enough to conclude that there is a significant linear relationship between the variables, or the value of \(r\) is due to chance.
To make this decision, you use a hypothesis-testing procedure. The traditional method is similar to the one used in previous chapters.
In hypothesis testing, one of these is true:
\(H_0:\) \(\rho=0\) This null hypothesis means that there is no correlation between the x and y variables in the population.
\(H_1:\) \(\rho\neq0\) This alternative hypothesis means that there is a significant correlation between the variables in the population.
Formula for the \(t\) test for the Correlation Coefficient
\[t=r\cdot\sqrt{\frac{n-2}{1-r^2}}\]
with degrees of freedom equal to \(n-2\), where \(n\) is the number of ordered pairs \(x,y\).