Confidence intervals and significance tests

A sample is used both to construct a 95% confidence interval for a population proportion p and to run a significance test with null hypothesis $H_0:P=0.07$ and significance level $\alpha=0.05$.

Is it possible that $p = 0.7$ falls outside of the 95% confidence itnerval, yet $H_0$ is not rejected?

I'm assuming that since a two-sided test at significance level $\alpha$ gives roughly the same result as a $100(1-\alpha)\%$ confidence interval, this is not possible.

Also, in terms of the sample size required for a significance test, does it just have to statisfy the normal condition - $np_0\geq10$ and $n(1-p_0)\geq10$?

Under normal circumstances, no. Once you fall outside the confidence interval, you are in what's called the Critical Region for which the alternative hypothesis applies. That said, Confidence intervals should not be used for hypothesis testing. Second, the condition of $np>10$ and $nq>10$ is used to ensure that the approximation from binomial to normal is "acceptable" The value of $10$ is kind of arbitrarily chosen. (The higher the better) Of course, depending on the sample, one would assume that binomial and normal are proper distributions for what you are testing.
• I did a bit more reading in my textbook, and I realized that the formula for the standard deviation of $\hat p$ used in the significance test uses the hypothesized value of $p$, whereas the formula for the standard error of $\hat p$ used in the construction of the confidence interval uses the sample value of $p$, so that the difference makes it possible for the sample proportion to be far enough away from 0.7 that 0.7 does not lie in the 95% confidence interval, but not far enough away that $H_0$ is rejected in the significance test. – Auguste Baudin Mar 23 '16 at 23:53
It is not possible by the definition of hypothesis test. Since $p=0.7$ falls outside of the 95% confidence interval and it is a 2-tails test, we need to reject $H_0$.