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Sagot :
To determine the correlation coefficient for the given data, follow these steps:
1. Understand the data points:
The data points given in the table are:
[tex]\[ \begin{array}{cc} x & y \\ \hline 0 & 0 \\ 1 & 1 \\ 4 & 4 \\ 5 & 5 \\ \end{array} \][/tex]
2. Correlation coefficient formula:
The correlation coefficient [tex]\( r \)[/tex] for two sets of data [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is calculated using the formula:
[tex]\[ r = \frac{ \sum (x_i - \bar{x})(y_i - \bar{y}) }{ \sqrt{\sum (x_i - \bar{x})^2} \sqrt{\sum (y_i - \bar{y})^2} } \][/tex]
Here, [tex]\( \bar{x} \)[/tex] and [tex]\( \bar{y} \)[/tex] are the means of the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] data sets, respectively.
3. Calculate the means of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ \bar{x} = \frac{0 + 1 + 4 + 5}{4} = 2.5 \][/tex]
[tex]\[ \bar{y} = \frac{0 + 1 + 4 + 5}{4} = 2.5 \][/tex]
4. Determine the differences from the mean:
For [tex]\( x \)[/tex]:
[tex]\[ x_i - \bar{x}: -2.5, -1.5, 1.5, 2.5 \][/tex]
For [tex]\( y \)[/tex]:
[tex]\[ y_i - \bar{y}: -2.5, -1.5, 1.5, 2.5 \][/tex]
5. Calculate the products of the differences [tex]\( (x_i - \bar{x})(y_i - \bar{y}) \)[/tex]:
[tex]\[ (-2.5)(-2.5) = 6.25 \][/tex]
[tex]\[ (-1.5)(-1.5) = 2.25 \][/tex]
[tex]\[ (1.5)(1.5) = 2.25 \][/tex]
[tex]\[ (2.5)(2.5) = 6.25 \][/tex]
[tex]\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = 6.25 + 2.25 + 2.25 + 6.25 = 17 \][/tex]
6. Calculate the squares of the differences:
For [tex]\( x \)[/tex]:
[tex]\[ (x_i - \bar{x})^2: 6.25, 2.25, 2.25, 6.25 \][/tex]
[tex]\[ \sum (x_i - \bar{x})^2 = 6.25 + 2.25 + 2.25 + 6.25 = 17 \][/tex]
For [tex]\( y \)[/tex]:
[tex]\[ (y_i - \bar{y})^2: 6.25, 2.25, 2.25, 6.25 \][/tex]
[tex]\[ \sum (y_i - \bar{y})^2 = 6.25 + 2.25 + 2.25 + 6.25 = 17 \][/tex]
7. Substitute these values into the correlation coefficient formula:
[tex]\[ r = \frac{17}{\sqrt{17} \cdot \sqrt{17}} = \frac{17}{17} = 1 \][/tex]
Given the data points and the calculations, the correlation coefficient for the given data is approximately 0.9999999999999998.
So, the correlation coefficient is very close to 1.
1. Understand the data points:
The data points given in the table are:
[tex]\[ \begin{array}{cc} x & y \\ \hline 0 & 0 \\ 1 & 1 \\ 4 & 4 \\ 5 & 5 \\ \end{array} \][/tex]
2. Correlation coefficient formula:
The correlation coefficient [tex]\( r \)[/tex] for two sets of data [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is calculated using the formula:
[tex]\[ r = \frac{ \sum (x_i - \bar{x})(y_i - \bar{y}) }{ \sqrt{\sum (x_i - \bar{x})^2} \sqrt{\sum (y_i - \bar{y})^2} } \][/tex]
Here, [tex]\( \bar{x} \)[/tex] and [tex]\( \bar{y} \)[/tex] are the means of the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] data sets, respectively.
3. Calculate the means of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ \bar{x} = \frac{0 + 1 + 4 + 5}{4} = 2.5 \][/tex]
[tex]\[ \bar{y} = \frac{0 + 1 + 4 + 5}{4} = 2.5 \][/tex]
4. Determine the differences from the mean:
For [tex]\( x \)[/tex]:
[tex]\[ x_i - \bar{x}: -2.5, -1.5, 1.5, 2.5 \][/tex]
For [tex]\( y \)[/tex]:
[tex]\[ y_i - \bar{y}: -2.5, -1.5, 1.5, 2.5 \][/tex]
5. Calculate the products of the differences [tex]\( (x_i - \bar{x})(y_i - \bar{y}) \)[/tex]:
[tex]\[ (-2.5)(-2.5) = 6.25 \][/tex]
[tex]\[ (-1.5)(-1.5) = 2.25 \][/tex]
[tex]\[ (1.5)(1.5) = 2.25 \][/tex]
[tex]\[ (2.5)(2.5) = 6.25 \][/tex]
[tex]\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = 6.25 + 2.25 + 2.25 + 6.25 = 17 \][/tex]
6. Calculate the squares of the differences:
For [tex]\( x \)[/tex]:
[tex]\[ (x_i - \bar{x})^2: 6.25, 2.25, 2.25, 6.25 \][/tex]
[tex]\[ \sum (x_i - \bar{x})^2 = 6.25 + 2.25 + 2.25 + 6.25 = 17 \][/tex]
For [tex]\( y \)[/tex]:
[tex]\[ (y_i - \bar{y})^2: 6.25, 2.25, 2.25, 6.25 \][/tex]
[tex]\[ \sum (y_i - \bar{y})^2 = 6.25 + 2.25 + 2.25 + 6.25 = 17 \][/tex]
7. Substitute these values into the correlation coefficient formula:
[tex]\[ r = \frac{17}{\sqrt{17} \cdot \sqrt{17}} = \frac{17}{17} = 1 \][/tex]
Given the data points and the calculations, the correlation coefficient for the given data is approximately 0.9999999999999998.
So, the correlation coefficient is very close to 1.
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