Absolutely, let's determine the correlation coefficient for the provided data set step by step.
Given the data set:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 15 \\
\hline
6 & 13 \\
\hline
7 & 9 \\
\hline
8 & 8 \\
\hline
12 & 5 \\
\hline
\end{array}
\][/tex]
The correlation coefficient measures the strength and direction of the linear relationship between two variables. The formula for the Pearson correlation coefficient [tex]\( r \)[/tex] is:
[tex]\[
r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}
\][/tex]
Where:
- [tex]\( n \)[/tex] is the number of data points.
- [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are the variables.
- [tex]\( \Sigma \)[/tex] denotes the summation.
Based on the provided dataset:
[tex]\[
x = [2, 6, 7, 8, 12]
\][/tex]
[tex]\[
y = [15, 13, 9, 8, 5]
\][/tex]
Let's use the appropriate calculations to find the correlation coefficient.
1. Calculate the sums:
[tex]\[
\Sigma x = 2 + 6 + 7 + 8 + 12 = 35
\][/tex]
[tex]\[
\Sigma y = 15 + 13 + 9 + 8 + 5 = 50
\][/tex]
[tex]\[
\Sigma xy = (215) + (613) + (79) + (88) + (12*5) = 30 + 78 + 63 + 64 + 60 = 295
\][/tex]
[tex]\[
\Sigma x^2 = (2^2) + (6^2) + (7^2) + (8^2) + (12^2) = 4 + 36 + 49 + 64 + 144 = 297
\][/tex]
[tex]\[
\Sigma y^2 = (15^2) + (13^2) + (9^2) + (8^2) + (5^2) = 225 + 169 + 81 + 64 + 25 = 564
\][/tex]
2. Substitute the sums into the formula:
[tex]\[
r = \frac{5(295) - (35)(50)}{\sqrt{[5(297) - (35)^2][5(564) - (50)^2]}}
\][/tex]
3. Simplify the numerator:
[tex]\[
5(295) - 35(50) = 1475 - 1750 = -275
\][/tex]
4. Simplify the denominator:
[tex]\[
5(297) - 35^2 = 1485 - 1225 = 260
\][/tex]
[tex]\[
5(564) - 50^2 = 2820 - 2500 = 320
\][/tex]
[tex]\[
\sqrt{260 \times 320} = \sqrt{83200} \approx 288.40
\][/tex]
5. Calculate the final value:
[tex]\[
r = \frac{-275}{288.40} \approx -0.953
\][/tex]
So, the correlation coefficient is approximately [tex]\( -0.953 \)[/tex].
Therefore, the correct answer is:
D. -0.953
