p-Values

The p-value is the probability, computed assuming the null hypothesis is true, of getting a statistic at least as extreme as the observed one in the direction the alternative specifies. Mechanically it is a tail area of the null sampling distribution: for $H_0: p = 0.40$, $n = 250$, and $\hat{p} = 0.352$, the null standard deviation is $\sqrt{\frac{0.40(0.60)}{250}} \approx 0.0310$, $z \approx -1.55$, and the left-tail area is about 0.061. A one-sided alternative counts one tail, a two-sided alternative counts both and so doubles the value to about 0.121 for the same data.

Every standard misreading is the definition with its conditional clause removed. A p-value of 0.061 becomes a 6.1% chance the null is true, or a 93.9% chance the alternative is, or the probability that the results are due to chance, though the calculation already assumes chance alone. It becomes the probability of exactly this sample, though the definition says at least as extreme. The repair is the same each time: state the assumed value of $p$, the direction, and the sample size, as in if $p$ really were 0.40, about 6% of samples of 250 would give 0.352 or lower.