Answer :
Certainly! To find the correlation coefficient between the density of population and death rate for the given cities, follow these detailed steps:
1. List the given data:
- Population density (\(x\)):
[tex]\[ 200, 500, 400, 700, 600, 300 \][/tex]
- Death rate (\(y\)):
[tex]\[ 10, 12, 10, 15, 9, 12 \][/tex]
2. Calculate the mean of the datasets:
- Mean density (\(\bar{x}\)):
[tex]\[ \bar{x} = \frac{200 + 500 + 400 + 700 + 600 + 300}{6} = \frac{2700}{6} = 450.0 \][/tex]
- Mean death rate (\(\bar{y}\)):
[tex]\[ \bar{y} = \frac{10 + 12 + 10 + 15 + 9 + 12}{6} \approx \frac{68}{6} \approx 11.3333 \][/tex]
3. Calculate the numerator for the correlation coefficient:
The numerator component is the sum of the product of the deviations of each pair of data points from their respective means:
[tex]\[ \sum (x_i - \bar{x})(y_i - \bar{y}) \][/tex]
Calculate the deviations and their products individually:
[tex]\[ \begin{aligned} & (200 - 450)(10 - 11.3333) \approx (200 - 450)(10 - 11.3333) = (-250)(-1.3333) \approx 333.33 \\ & (500 - 450)(12 - 11.3333) = (50)(0.6667) \approx 33.33 \\ & (400 - 450)(10 - 11.3333) = (-50)(-1.3333) \approx 66.67 \\ & (700 - 450)(15 - 11.3333) = (250)(3.6667) \approx 916.67 \\ & (600 - 450)(9 - 11.3333) = (150)(-2.3333) \approx -350.00 \\ & (300 - 450)(12 - 11.3333) = (-150)(0.6667) \approx -100.00 \\ \end{aligned} \][/tex]
Summing these products:
[tex]\[ 333.33 + 33.33 + 66.67 + 916.67 - 350.00 - 100.00 = 900.0 \][/tex]
4. Calculate the denominator for the correlation coefficient:
The denominator is the product of the square roots of the sum of squared deviations:
[tex]\[ \sqrt{\sum (x_i - \bar{x})^2} \times \sqrt{\sum (y_i - \bar{y})^2} \][/tex]
Calculate the squared deviations separately:
[tex]\[ \begin{aligned} & (200 - 450)^2 = 62500 \\ & (500 - 450)^2 = 2500 \\ & (400 - 450)^2 = 2500 \\ & (700 - 450)^2 = 62500 \\ & (600 - 450)^2 = 22500 \\ & (300 - 450)^2 = 22500 \\ \end{aligned} \][/tex]
Sum of the squared deviations for density (\(\sum (x_i - \bar{x})^2\)):
[tex]\[ 62500 + 2500 + 2500 + 62500 + 22500 + 22500 = 172000 \][/tex]
Calculate the squared deviations separately for death rate:
[tex]\[ \begin{aligned} & (10 - 11.3333)^2 \approx 1.7778 \\ & (12 - 11.3333)^2 \approx 0.4444 \\ & (10 - 11.3333)^2 \approx 1.7778 \\ & (15 - 11.3333)^2 \approx 13.4444 \\ & (9 - 11.3333)^2 \approx 5.4444 \\ & (12 - 11.3333)^2 \approx 0.4444 \\ \end{aligned} \][/tex]
Sum of the squared deviations for death rate (\(\sum (y_i - \bar{y})^2\)):
[tex]\[ 1.7778 + 0.4444 + 1.7778 + 13.4444 + 5.4444 + 0.4444 = 23.3333 \][/tex]
Product of the square roots:
[tex]\[ \sqrt{172000} \times \sqrt{23.3333} \approx 414.76 \times 4.83 \approx 2020.726 \][/tex]
5. Calculate the correlation coefficient (\(r\)):
Finally, use the correlation formula:
[tex]\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \][/tex]
Substituting the calculated values:
[tex]\[ r = \frac{900.0}{2020.726} \approx 0.445 \][/tex]
Therefore, the correlation coefficient between the density of population and the death rate for the given cities is approximately [tex]\(0.445\)[/tex].
1. List the given data:
- Population density (\(x\)):
[tex]\[ 200, 500, 400, 700, 600, 300 \][/tex]
- Death rate (\(y\)):
[tex]\[ 10, 12, 10, 15, 9, 12 \][/tex]
2. Calculate the mean of the datasets:
- Mean density (\(\bar{x}\)):
[tex]\[ \bar{x} = \frac{200 + 500 + 400 + 700 + 600 + 300}{6} = \frac{2700}{6} = 450.0 \][/tex]
- Mean death rate (\(\bar{y}\)):
[tex]\[ \bar{y} = \frac{10 + 12 + 10 + 15 + 9 + 12}{6} \approx \frac{68}{6} \approx 11.3333 \][/tex]
3. Calculate the numerator for the correlation coefficient:
The numerator component is the sum of the product of the deviations of each pair of data points from their respective means:
[tex]\[ \sum (x_i - \bar{x})(y_i - \bar{y}) \][/tex]
Calculate the deviations and their products individually:
[tex]\[ \begin{aligned} & (200 - 450)(10 - 11.3333) \approx (200 - 450)(10 - 11.3333) = (-250)(-1.3333) \approx 333.33 \\ & (500 - 450)(12 - 11.3333) = (50)(0.6667) \approx 33.33 \\ & (400 - 450)(10 - 11.3333) = (-50)(-1.3333) \approx 66.67 \\ & (700 - 450)(15 - 11.3333) = (250)(3.6667) \approx 916.67 \\ & (600 - 450)(9 - 11.3333) = (150)(-2.3333) \approx -350.00 \\ & (300 - 450)(12 - 11.3333) = (-150)(0.6667) \approx -100.00 \\ \end{aligned} \][/tex]
Summing these products:
[tex]\[ 333.33 + 33.33 + 66.67 + 916.67 - 350.00 - 100.00 = 900.0 \][/tex]
4. Calculate the denominator for the correlation coefficient:
The denominator is the product of the square roots of the sum of squared deviations:
[tex]\[ \sqrt{\sum (x_i - \bar{x})^2} \times \sqrt{\sum (y_i - \bar{y})^2} \][/tex]
Calculate the squared deviations separately:
[tex]\[ \begin{aligned} & (200 - 450)^2 = 62500 \\ & (500 - 450)^2 = 2500 \\ & (400 - 450)^2 = 2500 \\ & (700 - 450)^2 = 62500 \\ & (600 - 450)^2 = 22500 \\ & (300 - 450)^2 = 22500 \\ \end{aligned} \][/tex]
Sum of the squared deviations for density (\(\sum (x_i - \bar{x})^2\)):
[tex]\[ 62500 + 2500 + 2500 + 62500 + 22500 + 22500 = 172000 \][/tex]
Calculate the squared deviations separately for death rate:
[tex]\[ \begin{aligned} & (10 - 11.3333)^2 \approx 1.7778 \\ & (12 - 11.3333)^2 \approx 0.4444 \\ & (10 - 11.3333)^2 \approx 1.7778 \\ & (15 - 11.3333)^2 \approx 13.4444 \\ & (9 - 11.3333)^2 \approx 5.4444 \\ & (12 - 11.3333)^2 \approx 0.4444 \\ \end{aligned} \][/tex]
Sum of the squared deviations for death rate (\(\sum (y_i - \bar{y})^2\)):
[tex]\[ 1.7778 + 0.4444 + 1.7778 + 13.4444 + 5.4444 + 0.4444 = 23.3333 \][/tex]
Product of the square roots:
[tex]\[ \sqrt{172000} \times \sqrt{23.3333} \approx 414.76 \times 4.83 \approx 2020.726 \][/tex]
5. Calculate the correlation coefficient (\(r\)):
Finally, use the correlation formula:
[tex]\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \][/tex]
Substituting the calculated values:
[tex]\[ r = \frac{900.0}{2020.726} \approx 0.445 \][/tex]
Therefore, the correlation coefficient between the density of population and the death rate for the given cities is approximately [tex]\(0.445\)[/tex].
