Hypothesis test for mutually exclusive samples from a single population? Let's say I gave 100,000 people a survey on their favorite fruit, where they can bubble in "Apple" or "Pear" (but not both). They can also leave it blank.
The results are 90,000 people left it blank, 4,000 said "Apple", and 6,000 said "Pear".
How do I test whether the proportion of people who like apples is different from the proportion of people who like pears?
The standard 2-proportion z-test assumes the two samples are independent, but here they are not, since the more people fill in "Apple", the less people are able to fill in "Pear".
 A: You couldn't do a two sample test for equality of proportions, because you don't have two samples.  Thus, there is also no violation of the assumption of independence.  Instead, you have a single binomial with a lot of missing data.  If you feel comfortable with the assumption that the left blank responses are missing at random, you can conduct a simple binomial test with a null proportion of $.50$.  (If you are not comfortable assuming the left blank responses are missing at random, there won't be much you can do, because there is too much missing data—any outcome is possible.)  
Here is an example binomial test, coded in R:
binom.test(4000, 6000)
#   Exact binomial test
# 
# data:  4000 and 6000
# number of successes = 4000, number of trials =
# 6000, p-value < 2.2e-16
# alternative hypothesis: true probability of success is not equal to 0.5
# 95 percent confidence interval:
#  0.6545765 0.6785946
# sample estimates:
# probability of success 
#              0.6666667 

A: There are a few different options, which is best depends on what assumptions you are willing to make, what information you consider there to be in the "no answer" group, how you want to generalize, and other things like that.  Here are a few possibilities:
You are correct that the answers are not independent, but the covariance is known for cases like this so you can still construct an approximate z test that takes the covariance into account.
You can essentially do a McNemar's test using the "apple" and "pear" as the off-diagonal responses.  I think that this will reduce mathematically to be the same as the previous option.
If you condition on just those people who responded, then the only way for the 2 proportions to be the same is if they are $0.5$, so just do a 1 sample test on pears (or apples) against the 50% null hypothesis.
Fit a Bayesian binomial or multinomial model then look at the posterior distribution of the difference or ratio of the 2 proportions.
