Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
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].
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
