Chi-sq is supposed to be calculated $\sum_{i=1}^R \sum_{j=1}^C \frac{(O_{i,j} - e_{i,j})^2}{e_{i,j}}$. I thought there are two groups and it's even consistent with their chi-square estimate of which is $4, (R-1)(C-1) = 4*1$ Why is it that they don't compute it the same? I thought chi-square is for tables of all sorts. I know it seems like this one is odd because I would have tried calculating $e_{i,j} = \frac{(286 + 0.08x)x}{3858+x}$ where $x =$ the value of deaths in 1975.
SO what is going wrong? I'm stupid I know.
Although it's presented as a 5 $\times$ 2 table, it sounds like they're actually asking you to carry out a chi-squared Goodness of Fit test on the counts in the second column, using the first column as the hypothetical distribution under the null.
i.e. you have:
$$O_1 = 286, O_2 = 805, \dots, O_5=464,$$
and
$$e_1=0.08\times3858 = 308.64,\dots, e_5=0.11\times3858 = 424.38.$$
The appropriate chi-squared statistic is
$$\chi^2=\sum_{i=1}^5 \frac{(O_i - e_i)^2}{e_i} = \frac{(286 - 308.64)^2}{308.64}+\dots + \frac{(464 - 424.38)^2}{424.38}$$
with $5-1=4$ degrees of freedom.