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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).
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.
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.
