1. **Stating the problem:** We have a dataset of companies with their compensation in millions of dollars and their stock return percentages. We want to analyze the relationship between compensation and stock return. 2. **Formula and rules:** To analyze the relationship, we can calculate the correlation coefficient $r$ using the formula: $$ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}} $$ where $x$ is compensation, $y$ is stock return, and $n$ is the number of data points. 3. **Intermediate work:** - Calculate sums: $\sum x$, $\sum y$, $\sum xy$, $\sum x^2$, $\sum y^2$ - Use the data: - $\sum x = 12.8 + 5.6 + 8.1 + 11.3 + 16.9 + 20 + 7.2 + 7.7 + 9.6 + 14.7 + 2.6 + 4.2 = 120.3$ - $\sum y = -2.9 + 3.8 + 57 + 71 - 11 + 38 - 7.9 + 32 - 0.3 + 2.8 + 42 + 11 = 235.5$ - Calculate $\sum xy$: $12.8(-2.9) + 5.6(3.8) + 8.1(57) + 11.3(71) + 16.9(-11) + 20(38) + 7.2(-7.9) + 7.7(32) + 9.6(-0.3) + 14.7(2.8) + 2.6(42) + 4.2(11) = -37.12 + 21.28 + 461.7 + 802.3 - 185.9 + 760 - 56.88 + 246.4 - 2.88 + 41.16 + 109.2 + 46.2 = 2105.36$ - Calculate $\sum x^2$: $12.8^2 + 5.6^2 + 8.1^2 + 11.3^2 + 16.9^2 + 20^2 + 7.2^2 + 7.7^2 + 9.6^2 + 14.7^2 + 2.6^2 + 4.2^2 = 163.84 + 31.36 + 65.61 + 127.69 + 285.61 + 400 + 51.84 + 59.29 + 92.16 + 216.09 + 6.76 + 17.64 = 1518.69$ - Calculate $\sum y^2$: $(-2.9)^2 + 3.8^2 + 57^2 + 71^2 + (-11)^2 + 38^2 + (-7.9)^2 + 32^2 + (-0.3)^2 + 2.8^2 + 42^2 + 11^2 = 8.41 + 14.44 + 3249 + 5041 + 121 + 1444 + 62.41 + 1024 + 0.09 + 7.84 + 1764 + 121 = 12857.03$ 4. **Calculate correlation coefficient $r$:** $$ r = \frac{12(2105.36) - (120.3)(235.5)}{\sqrt{[12(1518.69) - (120.3)^2][12(12857.03) - (235.5)^2]}} $$ Calculate numerator: $$ 12 \times 2105.36 = 25264.32 $$ $$ 120.3 \times 235.5 = 28339.65 $$ $$ \text{Numerator} = 25264.32 - 28339.65 = -3075.33 $$ Calculate denominator parts: $$ 12 \times 1518.69 = 18224.28 $$ $$ (120.3)^2 = 14472.09 $$ $$ 12 \times 12857.03 = 154284.36 $$ $$ (235.5)^2 = 55440.25 $$ $$ \text{Denominator} = \sqrt{(18224.28 - 14472.09)(154284.36 - 55440.25)} = \sqrt{3752.19 \times 98844.11} = \sqrt{370927027.5} = 19256.5 $$ 5. **Final correlation coefficient:** $$ r = \frac{-3075.33}{19256.5} \approx -0.16 $$ 6. **Interpretation:** The correlation coefficient $r \approx -0.16$ indicates a very weak negative linear relationship between compensation and stock return in this dataset. **Answer:** The correlation coefficient between compensation and stock return is approximately $-0.16$ indicating a weak negative correlation.