Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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].
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.