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.
