It should be noted that the three hypothesis tests we have learned for testing the existence of a linear relationship — the t-test for \(H_{0} \colon \beta_{1}= 0\), the ANOVA F-test for \(H_{0} \colon \beta_{1} = 0\), and the t-test for \(H_{0} \colon \rho = 0\) — will always yield the same results. For example, when evaluating whether or not a linear relationship exists between a husband's age and his wife's age if we treat the husband's age ("HAge") as the response and the wife's age ("WAge") as the predictor, each test yields a P-value of 0.000... < 0.001 (Husband and Wife data):

And similarly, if we treat the wife's age ("WAge") as the response and the husband's age ("HAge") as the predictor, each test yields of P-value of 0.000... < 0.001:

Technically, then, it doesn't matter what test you use to obtain the P-value. You will always get the same P-value. But, you should report the results of the test that make sense for your particular situation:

• If one of the variables can be clearly identified as the response, report that you conducted a t-test or F-test results for testing \(H_{0} \colon \beta_{1} =0\). (Does it make sense to use x to predict y?)
• If it is not obvious which variable is the response, report that you conducted a t-test for testing \(H_{0} \colon \rho = 0\). (Does it only make sense to look for an association between x and y?)