In the previous chapter, we introduced hypothesis testing as a means to quantify the probability that a given hypothesis (such as that the two samples were from a single population) is true. We will use the same process to quantify the probability that a correlation exists in the wider population based on our sample.

First, we must formulate two hypotheses, a null hypothesis and an alternate hypothesis:

H₀ is the hypothesis that the population correlation is zero. In other words, our conservative view is that the measured correlation is purely due to chance sampling error.

H₁ is the alternative possibility that the population correlation is not zero. Notice that we don't specify the direction of the correlation, only that there is one. This means we are performing a two-tailed test.

The standard error of the sample r is given by:

This formula is only accurate when is close to zero (recall that the magnitude of r influences our confidence), but fortunately, this is exactly...