To calculate the correlation coefficient [tex]\(r\)[/tex] between the two variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex], let's follow these steps:
1. Identify the given data:
The respective `x` and `y` values are:
[tex]\[
x: [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
\][/tex]
[tex]\[
y: [13.92, 16.63, 19.24, 19.55, 21.56, 20.97, 26.08, 24.79, 28.4, 28.91, 30.12, 34.43, 33.84, 36.75]
\][/tex]
2. Calculate the correlation coefficient:
The correlation coefficient [tex]\(r\)[/tex] is calculated using the formula:
[tex]\[
r = \frac{\text{cov}(x, y)}{\sigma_x \sigma_y}
\][/tex]
where [tex]\(\text{cov}(x, y)\)[/tex] is the covariance of [tex]\(x\)[/tex] and [tex]\(y\)[/tex], and [tex]\(\sigma_x\)[/tex] and [tex]\(\sigma_y\)[/tex] are the standard deviations of [tex]\(x\)[/tex] and [tex]\(y\)[/tex], respectively.
3. Using statistical methods, we find that:
- The covariance [tex]\(\text{cov}(x, y)\)[/tex] between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] can be computed.
- The standard deviations [tex]\(\sigma_x\)[/tex] and [tex]\(\sigma_y\)[/tex] for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] can be computed using provided data.
- Using these computations, the correlation coefficient [tex]\(r\)[/tex] will be determined.
4. Compute the correlation coefficient matrix:
- The correlation coefficient matrix for variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is a 2x2 matrix where the value at position (0,1) and (1,0) is the correlation coefficient between [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
5. Extract the correlation coefficient:
The correlation coefficient between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is found to be approximately [tex]\(0.9877573914478209\)[/tex].
6. Round to three decimal places:
Finally, we round this correlation coefficient to three decimal places:
[tex]\[
r \approx 0.988
\][/tex]
Therefore, the correlation coefficient [tex]\(r\)[/tex] is:
[tex]\[
r = 0.988
\][/tex]
