Significance probabilities (or P values) have been used in place of hypothesis tests as a means of giving more information about the relationship between the data and the hypothesis than does a simple reject/don't reject decision. Virtually all elementary statistics texts cover the calculation of P values for hypotheses concerning the mean of a sample from a normal distribution. Both one-sided hypotheses and point-null hypotheses are usually discussed. But there is a third case which is intermediate to the one-sided and point-null cases, namely the interval hypothesis, which receives no coverage in elementary texts. We show here that P values for interval hypotheses exhibit a paradoxical sort of behavior which was actually visible in the one-sided and point-null cases, but which has been largely ignored in the past.