“The p-value indicates that the probability of having a sample result of 0.031 when the underlying population coefficient is 0 is about 4.89%. Because this p-value is less than 5%, the null hypothesis is rejected for the intercept. The slope coefficient is −0.732. The p-value indicates that the probability of having a sample result of −0.732 when the underlying population coefficient is 0 is about 12.85%. Because the p-value exceeds the 5% level of significance, the null hypothesis is not rejected for the slope coefficient” This was an explanation to a question about statistical significance. I understand p is the lowest level of significance at which a null hypothesis should be rejected so it chosen level of significance is greater than than this the coefficient is statistically significant. I get that. but the above explanation says “p-value indicates that the probability of having a sample result of 0.031 when the underlying population coefficient is 0 is about 4.89%.” If there is only a 4.89% chance of having a sample result of 0.031 then how can we reject the null hypothesis that the coefficient is 0? whereas the second sentence says the probability of having a sample result of -0.732 is 12.85% but we accept the null hyp of 0. if there is a higher chance that the sample result is 12.85 how can we accept the null hypothesis. just logically thinking through this is not making sense to me. if anyone could explain this I would be really grateful.
They’ve left out a VERY important part of the definition, which I have added. A p-value is the probability of obtaining a result at least as extreme (at least as contradictory to the null) as the observed one, assuming the null is true. This is different from saying probability of obtaining X if the null is true, because it includes X and values more extreme.
This isn’t really the definition of p-value, it’s a simplified way of telling beginners what to do with the p-value. The p-value is the probability of equally or more extreme results as the observed result assuming the null hypothesis is true. The definition regarding lowest level of alpha to reject H0 is more operational and incorrectly implies that alpha may be chosen based on the p-value.
Again, they incorrectly left out that the .0489 probability is for obtaining a value of .031 or more extreme assuming H0 is true. Basically the logic is this: if H0 is true, we have observed something that is part of a set of outcomes (as extreme or more extreme implies a set) that has probability p-value of occurring. The smaller the p-value, the more evidence we have to contradict the null. Therefore, is our strength of evidence in great enough disagreement with the null (reject H0 at a preselected significance level, alpha). Here, they’re saying since .0489 is smaller than .05, that there is too much disagreement between the observed data and the null hypothesis, therefore, they reject the null hypothesis.
Again, it’s the probability of obtaining a result as extreme as -.0732 or more extreme, assuming the coefficient is actually zero (H0 is true). Second, there is no “accepting” H0. Failing to reject H0 is what occurs here. The reason this occurs when p-value> alpha is because the p-value is suggesting that our observations disagree with H0, but not to a strong enough level where we outright reject H0 (the p-value isn’t small enough to suggest “large enough” disagreement of the observation and H0).
thank you very much for your detailed explanation. I will process this and get back to you if I have any questions. thanks again.
I get this concept confused also. So in a nutshell it’s basically the inverse logic of the typical t test, where if the t statistic is greater than the critical value at a stated level of significance then we reject the null, and if the p value of a test of a specific outcome is less than that same significance level % then we also reject the null. Both results indicate significance and a need to reject the null. Is that fair?
Just remember that a p-value is a tail probability. If you recall how to find tail probabilities, you calculate a standardized value (z, t, f), and you find the area under the curve from that score out. The larger a score is, the smaller the area under the curve in the tail. Hence, test stat goes up, p-value goes down. So, yes, you have it summarized (although, I wouldn’t say it’s “inverse” the logic, because these are perfectly congruent)!
quality knowledge. Must be a quant?
Nope. In fact, I haven’t been in Finance for about 4 years. I just have a decent amount of coursework/teaching experience in statistics (this may now be clear why I seem to be a bit hyperactive when it comes to stats 
).