# Chi-Square goodness of fit test

i got the following task (im gonna translate everything so forgive me any mistakes):

A hunter is hunting wolves and taking notes on how many time he misses a shot before killing the wolf. The table shows how many times he missed (shooting 100 ($$n$$) wolves)

$$\begin{array} {|r|r|}\hline \text{amount of missed shots} & 0 & 1 & 2 & >2 \\ \hline \text{empirical frequency} & 72 & 18 & 8 & 2 \\ \hline \end{array}$$

My job is to check if the amount of missed shots is geometrically distributed with $$p=0.7$$ for an level of significance of $$\alpha = 0.05$$

I wanted to use a $$\chi^2$$ goodness of fit test but i didnt know how to properly choose my $$H_o$$ and $$H_1$$

What i did so far is:

Calculating the probablities for the geometrical distribution: $$P(X=0) = 0.7 ; \\ P(X=1)= 0.21;\\ P(X=2)= 0.063;\\ P(X\gt2)=0.027;\\$$

I also read that for approximation with the $$\chi^2$$ distribution one has to make sure that $$np_i$$ $$\gt5$$$$i$$ so i summed up the last two probabilities to $$P(X\ge2)=0.09$$.

$$H_0$$: Amount of missed shot is geometrically distributed with $$p = 0.7$$
$$H_1$$: Amount of missed shot is not geometrically distributed with $$p = 0.7$$
$$\begin{array} {|r|r|}\hline \text{amount of missed shots} & 0 & 1 & 2 & >2 \\ \hline \text{empirical frequency} & 72 & 18 & 8 & 2 \\ \hline \text{expected frequency} & 70 & 21 & 6.3 & 2.7\\ \hline \end{array}$$
$$\chi^2 = \frac{(72-70)^2}{70} + \frac{(18-21)^2}{21} + \frac{(8-6.3)^2}{6.3} + \frac{(2-2.7)^2}{2.7} \approx 1.1259$$