I'm trying to understand why a basic undergraduate intro to statistics still uses the normal approximation to do hypothesis tests for population proportions. It used to be that the computations were much easier using a normal approximation, but this is no longer the case.
Technically speaking, one "should" use the hypergeometric distribution, but since the population size is usually extremely large and rarely known, it seems best in these cases to use a binomial distribution instead.
Example. Let's say we want to test $H_0:p=.46$ versus $H_a:p<.46$. We get a sample proportion of $559/1267$. Using the normal approximation (in particular, using 1-PropZTest in the TI-84) we find that the P-value is $$P\left(\left.\hat{p}\leq\frac{559}{1267}\right|p=.46\right)\approx.0897$$ This is not hard to compute with the binomcdf function on the TI-84. This is because $$P\left(\left.\hat{p}\leq\frac{559}{1267}\right|p=.46\right)\approx P(x\leq 559)$$ where $x$ is the number of "successes" in 1267 "trials" of a binomial distribution with $p=.46$. Using binomcdf on the TI-84 we obtain a P-value of approximately $.0942$. This is not an insignificant difference. The relative error here is about $-4.8\%$.
If we use continuity correction with the normal distribution then we obtain $.0943$ which is a lot better of an approximation, but this is just as difficult if not more so to compute than the even more accurate binomial approximation.
Question 1. Are there any "real-world" circumstances under which it is appropriate to use a normal approximation without continuity correction?
It seems to me that the answer must be "no," but I am to instruct students with a textbook (Sullivan) that uses normal approximation without continuity correction for hypothesis testing. In fact, it gets worse because Sullivan rounds midway through his calculations which increases the error. His answer to the above was $.0869$, inflating the relative error to a whopping $-7.8\%$!
Question 2. Is this a serious flaw in the Sullivan textbook, or should I just live with such approximations?
Now, we can certainly use the binomial distribution for a 1-tailed test, as the P-value in that case has a clear interpretation as above. P-values for symmetric distributions in the two-tailed case are also natural. But the binomial distribution is nonsymmetric.
Question 3. How do we interpret the P-value for a two-tailed proportion test using the binomial distribution?
And similarly:
Question 4. How do we interpret a confidence interval for a binomial distribution?
Thanks in advance!