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Sagot :
Sure! Let's calculate the correlation coefficient for the given data step-by-step.
First, we need to understand that the correlation coefficient, often denoted as [tex]\( r \)[/tex], measures the strength and direction of the linear relationship between two variables. The formula for the Pearson correlation coefficient [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \][/tex]
Where:
- [tex]\( n \)[/tex] is the number of data points,
- [tex]\( \sum xy \)[/tex] is the sum of the product of each pair of [tex]\( x \)[/tex] and [tex]\( y \)[/tex],
- [tex]\( \sum x \)[/tex] is the sum of the [tex]\( x \)[/tex]-values,
- [tex]\( \sum y \)[/tex] is the sum of the [tex]\( y \)[/tex]-values,
- [tex]\( \sum x^2 \)[/tex] is the sum of the squares of the [tex]\( x \)[/tex]-values,
- [tex]\( \sum y^2 \)[/tex] is the sum of the squares of the [tex]\( y \)[/tex]-values.
Given the data:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 15 \\ \hline 5 & 10 \\ \hline 10 & 5 \\ \hline 15 & 0 \\ \hline \end{array} \][/tex]
Let's find each part of the formula:
1. Calculate the sums:
[tex]\[ \sum x = 0 + 5 + 10 + 15 = 30 \][/tex]
[tex]\[ \sum y = 15 + 10 + 5 + 0 = 30 \][/tex]
2. Calculate the sum of the products [tex]\( xy \)[/tex]:
[tex]\[ \sum xy = (0 \cdot 15) + (5 \cdot 10) + (10 \cdot 5) + (15 \cdot 0) = 0 + 50 + 50 + 0 = 100 \][/tex]
3. Calculate the sum of the squares [tex]\( x^2 \)[/tex]:
[tex]\[ \sum x^2 = 0^2 + 5^2 + 10^2 + 15^2 = 0 + 25 + 100 + 225 = 350 \][/tex]
4. Calculate the sum of the squares [tex]\( y^2 \)[/tex]:
[tex]\[ \sum y^2 = 15^2 + 10^2 + 5^2 + 0^2 = 225 + 100 + 25 + 0 = 350 \][/tex]
Now, substitute these sums into the formula for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{4(100) - (30)(30)}{\sqrt{[4(350) - 30^2][4(350) - 30^2]}} \][/tex]
Simplify the numerator:
[tex]\[ r = \frac{400 - 900}{\sqrt{[1400 - 900][1400 - 900]}} \][/tex]
Simplify the denominator:
[tex]\[ r = \frac{-500}{\sqrt{500 \times 500}} \][/tex]
[tex]\[ r = \frac{-500}{500} \][/tex]
[tex]\[ r = -1 \][/tex]
Thus, the correlation coefficient [tex]\( r \)[/tex] for the given data is [tex]\( -1 \)[/tex].
This indicates a perfect negative linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. The answer is:
[tex]\[ \boxed{-1} \][/tex]
First, we need to understand that the correlation coefficient, often denoted as [tex]\( r \)[/tex], measures the strength and direction of the linear relationship between two variables. The formula for the Pearson correlation coefficient [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \][/tex]
Where:
- [tex]\( n \)[/tex] is the number of data points,
- [tex]\( \sum xy \)[/tex] is the sum of the product of each pair of [tex]\( x \)[/tex] and [tex]\( y \)[/tex],
- [tex]\( \sum x \)[/tex] is the sum of the [tex]\( x \)[/tex]-values,
- [tex]\( \sum y \)[/tex] is the sum of the [tex]\( y \)[/tex]-values,
- [tex]\( \sum x^2 \)[/tex] is the sum of the squares of the [tex]\( x \)[/tex]-values,
- [tex]\( \sum y^2 \)[/tex] is the sum of the squares of the [tex]\( y \)[/tex]-values.
Given the data:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 15 \\ \hline 5 & 10 \\ \hline 10 & 5 \\ \hline 15 & 0 \\ \hline \end{array} \][/tex]
Let's find each part of the formula:
1. Calculate the sums:
[tex]\[ \sum x = 0 + 5 + 10 + 15 = 30 \][/tex]
[tex]\[ \sum y = 15 + 10 + 5 + 0 = 30 \][/tex]
2. Calculate the sum of the products [tex]\( xy \)[/tex]:
[tex]\[ \sum xy = (0 \cdot 15) + (5 \cdot 10) + (10 \cdot 5) + (15 \cdot 0) = 0 + 50 + 50 + 0 = 100 \][/tex]
3. Calculate the sum of the squares [tex]\( x^2 \)[/tex]:
[tex]\[ \sum x^2 = 0^2 + 5^2 + 10^2 + 15^2 = 0 + 25 + 100 + 225 = 350 \][/tex]
4. Calculate the sum of the squares [tex]\( y^2 \)[/tex]:
[tex]\[ \sum y^2 = 15^2 + 10^2 + 5^2 + 0^2 = 225 + 100 + 25 + 0 = 350 \][/tex]
Now, substitute these sums into the formula for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{4(100) - (30)(30)}{\sqrt{[4(350) - 30^2][4(350) - 30^2]}} \][/tex]
Simplify the numerator:
[tex]\[ r = \frac{400 - 900}{\sqrt{[1400 - 900][1400 - 900]}} \][/tex]
Simplify the denominator:
[tex]\[ r = \frac{-500}{\sqrt{500 \times 500}} \][/tex]
[tex]\[ r = \frac{-500}{500} \][/tex]
[tex]\[ r = -1 \][/tex]
Thus, the correlation coefficient [tex]\( r \)[/tex] for the given data is [tex]\( -1 \)[/tex].
This indicates a perfect negative linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. The answer is:
[tex]\[ \boxed{-1} \][/tex]
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