I know how to generate Pearson's $r$ correlation values in Excel and in R.
I understand the meaning of the value as it ranges from $-1<r<1$
I also understand hypothesis tests, confidence intervals, and $p$-value. ($p$ = the probability that this outcome was due to random chance or natural variation, and null hypothesis is true)
However, I am having trouble making the connection between Pearson's $r$ correlation and $p$-value. In a hypothesis test, there is some element of chance of the variation of outcomes (like coin flips). But, in a regression, it is based on actual data points. So what does p-value mean in this context? The odds that the data points are just clustered due to random chance? Is this directly based on sample size? I ask because I wonder how would a formula know anything about the variability of discrete data points (such as selling price and mileage of a car).
So, for an r value, the p-value is based on sample size? What I don't understand is that p-value seems to answer a binary question: Is there an effect or not? But for a correlation coefficient, there isn't a yes/no question being asked.
If I get an r = .8, and p-value of .20, what does that mean? It means there is a 20% chance that the correlation of .8 is not true? But, then what IS true? r=.7 ? r=.6?
Or is the rule of thumb that you can only use the r value, regardless of number, if p < .05 ?
If I get an r = .1, and p-value of .0001, what does that mean? We are very confident there is a very weak correlation? (How ironic?)
If I get an r = .5, and p-value of .0001, what does that mean? We are very confident there is a moderate correlation?
If I get an r = .5, and p-value of .3, what does that mean? It means there is a 30% chance that there is a moderate correlation of .5 ?