Link between p-values and confidence intervals

In Chapter 3 we developed the concepts of both the confidence interval (Cl) and the p-value. At that stage these ideas were kept separate. There is in fact a close link between the two and in this section we will develop that link. 

Note that the null hypothesis could be re-written Hq jXj — 1X2 = 0. 

The 95 per cent confidence interval a, b) for the difference in the treatment means, — p2 provides a range of plausible values for the true treatment difference. With 95 per cent confidence we can say that pj — p2 is somewhere within 

Statistical Thinking for Non-Statisticians in Drug Regulation Richard Kay 2007 John Wiley Sons, Ltd ISBN 978-0-470-31971-0 

In fact the link between the p-value and the confidence interval is not just operating at this informal level, it is much stronger and there is a mathematical connection between the two which goes as follows  

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