Post-hoc test for chi squared goodness-of-fit test: understanding Bonferroni I have seen this, but it doesn't help me here.
Here is my setup:
I have four categories, and want to simultaneously test whether p_1 = 0.4, p_2 = 0.35, p_3 = 0.15, p_4 = 0.10. I am using a chi squared goodness-of-fit test here.
I find that I should reject the null hypothesis, meaning that at least one of these population proportions is indeed statistically significantly different from its hypothesized value.
Then, post-hoc, I want to find which one(s) is/are significantly different. To do this, I've used a confidence interval on each population proportion, but found that all of the confidence intervals contain the hypothesized value - so that means I shouldn't have been able to reject the original null hypothesis!
The formula I used for my confidence intervals was:
$$\hat{p}_i \pm z_{0.00625}\sqrt{\frac{\hat{p}_i(1-\hat{p}_i)}{n}} $$
where $\hat{p}_i = $ the sample proportion for category $i$, $n = $ the sample size, and $z_{0.00625}$ is the Bonferroni-adjusted value from the normal distribution. From a table I found, this is 2.50.
I think I must've done something wrong here, but I can't figure out what. I've looked at this source and I've also read this SE question, but I can't seem to make the connection to my case.

*

*Statistical significance of the joint test does not imply statistical significance of one of the four individual tests for the proportions (see, e.g., this question).

*Your Bonferroni correction seems fine and it ensures (in your case asymptotically) strong control of the FWER. But it doesn't ensure a low type II error rate. One more powerful alternative would be the Holm method.

*The denominator of the radicand should be $n_i$ in case $n_i \neq n$

*$z_{0.00625} \approx  -2.5$.

*There are confidence intervals for binomial proportions, such as the Blyth-Still or the Agresti–Coull confidence interval, that are preferable to a Wald confidence interval.

