Properties of p-value

P-values for testing whether a linear model's slope is zero have the same properties as p-values for other hypothesis tests. In particular,

When Y and X are not related (β1 = 0), it is still possible to get small p-values, suggesting that β1 is not zero. However there is only probability 0.01 of getting a p-value as low as 0.01 — it is unlikely but possible. Such a p-value is more likely if the variables are related so we interpret it as giving strong evidence of a relationship.