Assess whether a data set is Poisson using a variance test As the title says, i'm not sure how a variance test determines that a data set is Poisson.
I've attempted the question and have come up with the following:
> c <-c(19, 17, 12, 16, 25, 25, 14, 14, 22, 14, 13, 18,17, 15, 16, 26, 17, 
18, 20, 12)

var(c)
[1] 18.05263

mean(c)
[1] 17.5

How does a variance test show that a data set could be Poisson?
 A: 
I'm not sure how a variance test determines that a data set is Poisson.

Nothing can really say that the data are drawn from a Poisson distribution, so the short answer is "it doesn't".

How does a variance test show that a data set could be Poisson?

A very different question to the previous one.
What the closeness of the mean and the variance may tell you is that the data set is not inconsistent with having come from a Poisson distribution.
There are various ways, formal and informal, that you could try to see whather this is consistent.
An informal method would be to simulate data from Poisson distributions with means in the vicinity of 17.5 (I'd be inclined to investigate cases from say 10 to 30 or so) and see either how variable the sample mean and variance are (look at a plot) or how often the ratio of sample variance to sample mean differs from 1 by at least as much as you saw here at each value of $\mu$ you consider. (It will happen quite often.)
An even more informal one would be to simply recognize that the variation in the sample variance would be expected to be reasonably large -- more than several units with a parameter in this region and so considerable deviation might be expected -- but this requires some degree of experience with Poisson samples.
More formally you might choose a reasonable test statistic (perhaps that ratio mentioned just above) and investigate the properties of that.
A few quick simulations suggest that at your sample size the ratio (var/mean) is very close to the same shape with mean and variance that are pretty stable across a range of $\lambda$ values (I tried values of the Poisson parameter between 10 and 30), so it looks like it would be possible to obtain a good hypothesis test quite easily. A normal approximation to the cube root (or a gamma approximation to the ratio) would probably work quite well.
There are other potential test statistics; don't feel constrained by that one.
