Karl Pearson's Method

To calculate the mean density of population, we use the formula: $ \text{Mean density} = \frac{\text{Total population}}{\text{Total area}} $. Substituting the given values, we get: $ \text{Mean density} = \frac{30+90+40+60+42+120}{150+180+100} = \frac{382}{430} \approx 0.888$ thousand persons per square mile.

The mean death rate can be calculated using the formula: $ \text{Mean death rate} = \frac{\text{Total number of deaths}}{\text{Total population}} \times 1000 $. Substituting the given values, we get: $ \text{Mean death rate} = \frac{300+1440+560+1224+312}{30+90+40+60+42} \times 1000 = \frac{3836}{262} \approx 14.61$ per 1000 population.

To calculate the coefficient of correlation by Karl Pearson's method, we use the formula: $ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} $. Substituting the given values, where $n$ is the number of data pairs, $\sum xy$ is the sum of the products of the paired values, $\sum x$ and $\sum y$ are the sum of the individual values, $\sum x^2$ and $\sum y^2$ are the sum of the squares of the individual values, we can calculate the coefficient of correlation.