Test if sample is Poisson distributed. Question regarding classes in chi square test

Question

I have to test if the following sample is poisson distributed

The numbers and their frequencies are:

$0\ 1\ 2\ 3\ 4\ 5$

$7\ 6\ 7\ 3\ 1\ 1$

I thought about using a chi square test with the clases $0$, $1$, $2$, $\geq 3$

The MLE is $$\hat \lambda :=\frac{0\cdot7+1\cdot6+2\cdot 7\dots+5\cdot 1}{25}=1.52$$

For $X\sim Poi(\hat \lambda)$ I receive $\mathbb P[X\geq 3]=0.196$

So for the expected number of observations for the class $\geq 3$ I receive $4.905$.

The problem is that we introduced a rule of thumb wich states the expected number should always be $\geq 5$. Is that a problem if it is only $4.905$? (For the other classes there is no problem because it is $\geq 5$). Should I use the classes $0, 1, \geq 2$ instead of $0, 1, 2, \ge 3$?

Answer 1

You are almost there.

First of all observe that your obsevation become more rare when they increase...thus it makes sense that your random sample actually comes form a poisson distribution.

then divide your sample into 4 classes: $0;1;2;\geq 3$ and compare your result with an expected poisson $Po(1.52)$ and do a chi square test with 2 degree of freedom...because you lost one degree using the estimation $\hat{\lambda}$

this is the test

the critical value at 5% is 5.99 thus you cannot reject the hypothesis that your random sample is from a poisson $Po(1.52)$