# What is the relationship between a p-value and a confidence interval?

Following from this question on the difference between confidence intervals and hypothesis testing, I would love to have a simple example to better understand the the relationship between confidence intervals and p-values.

For example, if I have $r=.8768 (n=300)$, then it is significant at $0.05$.

The confidence interval is (lower) $0.847729 < (r) 0.8768 < 0.900619$ (higher).

This is against 0 (i.e. no relationship).

• How does it rate with the confidence level testing?
• What do I look for?

## 2 Answers

The p-value relates to a test against the null hypothesis, usually that the parameter value is zero (no relationship). The wider the confidence interval on a parameter estimate is, the closer one of its extreme points will be to zero, and a p-value of 0.05 means that the 95% confidence interval just touches zero. In fact for a p-value $p$ of a parameter estimate, the $(1-p)$ level confidence interval just touches zero.

• Can you please explain this further. – Adhesh Josh Oct 7 '11 at 19:58

A $(1−\alpha)$ level confidence interval is exactly the range of values that would not be rejected using an $\alpha$ level test, assuming the same general theory generated the confidence interval and the hypothesis test decision. Therefore, they are exactly equivalent in terms of the decision about whether or not to reject the null hypothesis.

• When you say "the overall decision", I think you mean "a decision to reject the null hypothesis or not". There are other decisions, for example involving a quantative loss function, which might be made using a confidence interval but not a $p$ value. – Henry Oct 4 '11 at 13:22
• True. I've edited to reflect what you've said - thanks – Macro Oct 4 '11 at 13:23
• +1 @Henry There are things contained in a confidence interval not contained in a p-value. p-values (or worse, phrases like "significant at alpha = 0.05" are nothing more than exercises in lost information. – Fomite Oct 4 '11 at 18:12