Question on basic hypothesis testing I have a question about hypothesis testing.
How can I test a hypothesis like the following for example:
People prefer chicken to beef.
And for example, the data that I have is that I've asked from 1000 people "do they prefer beef or do they prefer chicken?" 80% said chicken, and 5% said beef and 15% indifferent.
In general, how can I test some hypothesis like that, without putting a limit on it (like more than 60% of the people prefer chicken to beef)?
 A: "People prefer chicken to beef" is not quite precise, so we need to 'unpack' the concept a bit:
"The proportion of people who prefer chicken to beef is larger than the proportion of people who prefer beef to chicken"
If that's what you really mean, that's something we can work with. 
You also have a sample which -- if it's a random sample from the population of interest -- yields a set of counts. Here's an example using your numbers:
  Prefer                    Prefer  |   
  Chicken    Indifferent    Beef    |   Total
                                    |        
    800         150          50     |   1000

The statistical question then is: if there's no preference of chicken to beef in the population of interest, could we observe such a sample outcome by chance?
The obvious model here is a multinomial model. We could condition on the people who express a preference (so there's an implied but uncontroversial "among those who express a preference" inserted into our hypotheses), yielding the relevant subtable:
  Prefer       Prefer  |   Total with
  Chicken      Beef    |   a preference
                       |
    800         50     |    850

This is now a straight binomial proportions test.
The original (bolded) research hypothesis in your question was framed as a one sided hypothesis (the two sided hypothesis would be "there's a preference for one or the other - they're not equally preferred"). 
However, you also need to ask yourself whether you really want to conduct the test using a one-sided hypothesis test or whether it's better to conduct a two-sided test and deal with the original one-sided hypothesis in the conclusion.
Since that's a different issue to the central one here (and addressed in other questions on site), I'll leave the issue aside for now -- and simply assume you do want a one-tailed test. Different authors present one-tailed tests somewhat differently but in this case because of the clean phrasing in your statement, I think the development is straightforward.
The null and alternative hypothesis in words is:
$H_0: \text{The proportion of people who prefer chicken to beef is not higher}\\ \qquad\text{than the proportion of people who prefer beef to chicken}$
$H_1: \text{The proportion of people who prefer chicken to beef is higher}\\ 
\qquad\text{than the proportion of people who prefer beef to chicken}$
and in terms of population parameters:
$H_0: \pi_C \leq \pi_B$
$H_1: \pi_C > \pi_B$
where $\pi_C$ and $\pi_B$ are the population proportions (among those expressing a preference) who prefer chicken and beef respectively.
But since we're conditioning on those expressing a preference (only considering two outcomes now), this is equivalent to:
$H_0: \pi_C \leq 0.5$
$H_1: \pi_C > 0.5$
And now this is a standard one-tailed binomial proportion test.
(Alternatively we could keep the original trinomial and formulate the hypothesis as a contrast. It would yield the same result.)
The sample size is such that we could just do a Z-test, but I'll go ahead and do it as a binomial test.
Under the null, the highest probability of observing a result as or more extreme (in the direction of the alternative) will be given by the equality case, so we work out our p-value by taking $\pi_C=0.5$. Let $X_C$ be the number in a sample of 850 (who express a preference) preferring chicken.
Then under that "best case", we have $X_C\sim\text{binomial}(850,0.5)$. The p-value is then $P(X_C\geq 800)$ (an astronomically tiny number which will lead you to reject the null hypothesis at any of the usual significance level choices; $p\approx 1.3 \times 10^{-256}$ -- however, the actual number is effectively meaningless, since even tiny deviations from the assumptions - which will certainly be the case (at least) - can change it by several orders of magnitude).
[Alternatively if you want to work strictly with a rejection rule, the 5% critical value for $X_C$ is $450$, and the 1% critical value is $460$, and you reject $H_0$ if you equal or exceed your critical value.]
If you did it using a Z-test, then with continuity correction your Z-statistic is $29.12$.
A: In hypothesis testing, you come up with...a hypothesis! So if the statement was 

People prefer chicken to beef

Maybe you try to quantify it by saying, "hmmm, I think over 50% of people prefer chicken to beef". It doesn't actually matter if people say they are indifferent, after all you walk into the experiment with no prior knowledge. 
In this situation you would perform a (binomial) hypothesis test which says $H_0:$ people are indifferent to chicken, whilst $H_1$: people would prefer chicken (one sided test). In this scenario you will have to have some kind of number that you're testing against. 
Or maybe, the survey to be location orientated and we have change our hypothesis to:

More people in country A list their favourite food as chicken than any other country.

In this case we can have a hypothesis which does not have some notion of a predetermined number/proportion, (this can probably be framed as a chi-squared test). 
So in general, asking or framing the question/hypothesis in the appropriate way can lead to different approaches and hence difference assumptions. There is no "one way" to do this, and learning to ask the best question is often the hardest part of statistics. 
