Sure! To calculate the correlation coefficient for the given data, you can follow these detailed steps:
1. Organize the Data:
The data given is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
4 & 93.6 \\
\hline
7 & 87.3 \\
\hline
9 & 73.5 \\
\hline
11 & 67.1 \\
\hline
13 & 90.9 \\
\hline
15 & 55.7 \\
\hline
19 & 25.5 \\
\hline
\end{array}
\][/tex]
2. Calculate the Means of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
\text{Mean of } x, \bar{x} = \frac{4 + 7 + 9 + 11 + 13 + 15 + 19}{7} = \frac{78}{7} \approx 11.14
\][/tex]
[tex]\[
\text{Mean of } y, \bar{y} = \frac{93.6 + 87.3 + 73.5 + 67.1 + 90.9 + 55.7 + 25.5}{7} = \frac{493.6}{7} \approx 70.51
\][/tex]
3. Compute the Differences from the Mean:
[tex]\[
x_i - \bar{x}, y_i - \bar{y}
\][/tex]
For example, for the first pair [tex]\((4, 93.6)\)[/tex]:
[tex]\[
4 - 11.14 \approx -7.14, \quad 93.6 - 70.51 \approx 23.09
\][/tex]
Repeat this for all data points.
4. Compute the Products of the Differences:
[tex]\[
(x_i - \bar{x})(y_i - \bar{y})
\][/tex]
Again, for the first pair:
[tex]\[
(-7.14)(23.09) \approx -164.89
\][/tex]
Repeat this for all pairs and find the sum of these products.
5. Sum of Squares for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
(x_i - \bar{x})^2, \quad (y_i - \bar{y})^2
\][/tex]
For the first [tex]\(x\)[/tex]:
[tex]\[
(-7.14)^2 \approx 50.98
\][/tex]
Repeat for all [tex]\(x\)[/tex] and find the sum of these squares. Do the same for [tex]\(y\)[/tex].
6. Calculate the Correlation Coefficient ([tex]\( r \)[/tex]):
The formula for the correlation coefficient [tex]\( r \)[/tex] is:
[tex]\[
r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2 \sum_{i=1}^{n} (y_i - \bar{y})^2}}
\][/tex]
Plug in the sums calculated from previous steps.
7. Round the Result to the Nearest Thousandth:
Using the calculations above, the correlation coefficient [tex]\( r \)[/tex] value you obtain is approximately:
[tex]\[
r \approx -0.839
\][/tex]
Therefore, the correlation coefficient for the given data, rounded to the nearest thousandth, is [tex]\( -0.839 \)[/tex].
