 Solution

After pooling, there are eight categories: $0,1,\dots ,6,\ge 7$. Their frequencies have one constraint because they sum to $N$, and one model parameter was fitted. Thus the reference chi-squared distribution hasdegrees of freedom. The reported statistic satisfies $25.914>22.46$, the stated upper $0.001$ cutoff for six degrees of freedom. Therefore reject the Poisson model at the 0.1% significance level under the independence and common-mean assumptions. The discrepancy is especially pronounced in the zero-count class, but the test assesses the whole fitted distribution. The conclusion uses the approximate Pearson chi-squared goodness-of-fit test, not an exact finite-sample tail probability.