what statistic would you use if you had

To analyze the relationship between an independent variable (geographic location) and a dependent variable (satisfaction with the living environment), we can use a statistical test called the ${\chi }^2$ test for independence.
The ${\chi }^2$ test for independence is used to determine if there is a significant association between two categorical variables. In this case, the independent variable (geographic location) has four categories (north, south, east, west), and the dependent variable (satisfaction with the living environment) has two categories (yes or no).
The null hypothesis for the ${\chi }^2$ test for independence states that there is no association between the two variables. The alternative hypothesis suggests that there is an association between the variables.
To perform the ${\chi }^2$ test for independence, we would create a contingency table that shows the frequency or count of observations for each combination of the categories of the two variables. It would look something like this:
$\begin{array}{ccc}\hfill \hfill & \hfill \text{Satisfaction: Yes}\hfill & \hfill \text{Satisfaction: No}\hfill \\ \hfill \text{North}\hfill & \hfill \text{Frequency}\hfill & \hfill \text{Frequency}\hfill \\ \hfill \text{South}\hfill & \hfill \text{Frequency}\hfill & \hfill \text{Frequency}\hfill \\ \hfill \text{East}\hfill & \hfill \text{Frequency}\hfill & \hfill \text{Frequency}\hfill \\ \hfill \text{West}\hfill & \hfill \text{Frequency}\hfill & \hfill \text{Frequency}\hfill \end{array}$
The observed frequencies in each cell of the contingency table would be based on the actual data collected. We would then calculate the expected frequencies under the assumption of independence between the variables.
Next, we would calculate the ${\chi }^2$ statistic, which measures the difference between the observed and expected frequencies, taking into account the sample size. The formula for calculating the ${\chi }^2$ statistic involves summing the squared differences between observed and expected frequencies, divided by the expected frequencies:
${\chi }^2=\sum \frac{(O_{ij}-E_{ij}{)}^2}{E_{ij}}$
where $O_{ij}$ represents the observed frequency in cell $(i,j)$ and $E_{ij}$ represents the expected frequency in cell $(i,j)$.
Finally, we would compare the calculated ${\chi }^2$ statistic to the critical value from the ${\chi }^2$ distribution with appropriate degrees of freedom. If the calculated ${\chi }^2$ value is greater than the critical value, we would reject the null hypothesis and conclude that there is a significant association between geographic location and satisfaction with the living environment.
Therefore, in this scenario, the appropriate statistical test would be the ${\chi }^2$ test for independence.