Can a confidence interval have boundaries that are different in size in relation to the mean? I saw a poll for local elections where 9.3% of the respondents said that they would vote for John Doe. The poll put the lower bound at 8.3% and the upper bound at 11.3%. From, what I have learned from introductory stats classes should not the distance of the mean to the lower bound and the distance of the upper bound be the same given that the formula is xbar plus or minus z(s/sqrt(n)). Does this poll have an error or am I missing something?
 A: Suppose that the poll used $n = 2400$ subjects of whom $X =223$
favored candidate Doe.
Then a traditional Wald confidence interval would start with a
point estimate $\hat p = 223/2400 = 0.093$ of the proportion in favor and the a 95% CI would be
$$\hat p \pm 1.96\sqrt{\frac{\hat p(1-\hat p)}{n}},$$
which computes to $(0.081,0.104)$ or $(8.1\%, 10.4\%).$
Notice that $\pm 1.96$ cuts 2.5% from each tail of a
standard normal distribution. This CI is sometimes called probability symmetric because
the same probability is cut from each tail of the distribution.
It is by far the most common type of CI.
x = 223;  n = 2400
p.hat = x/n;  q = qnorm(c(.025,.975))
p.hat; q
[1] 0.09291667
[1] -1.959964  1.959964
CI = p.hat + q*sqrt((p.hat *(1-p.hat))/n);  CI
[1] 0.08130184 0.10453150

However, one says a 95% CI because many kinds of CIs are possible For example, one can cut 4%
from one tail and 1% from the other and still have a 95% CI.
Then another 95% CI is $(0.083, 0.107)$ or $(8.3\%,10.7\%).$
For this type of CI, the point estimate is no longer at the center of the CI.
Especially for small percentages, such alternate types of CIs
may be used. (If for no other reason. this might be done to summarize a campaign poll in a way to encourage a candidate
who is not showing well in the polls.)
x = 223;  n = 2400
p.hat = x/n;  q = qnorm(c(.04,.99))
p.hat; q
[1] 0.09291667
[1] -1.750686  2.326348  
CI = p.hat + q*sqrt((p.hat *(1-p.hat))/n);  CI
[1] 0.08254203 0.10670270

So either the newspaper reporting the poll had a typographical error or the pollster is not using a probability-symmetric style
of CI. (I'm betting on the typo.)
