data table\nearnings per share, x: 1.00, 4.08, 3.45, 8.29, 1.6, 2.9\ndividends per share, y: 0.94, 0.46…

Explanation:

Step1: Recall correlation - coefficient formula

The sample correlation coefficient $r$ is given by the formula $r=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{\sqrt{[n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}][n\sum_{i = 1}^{n}y_i^{2}-(\sum_{i = 1}^{n}y_i)^{2}]}}$. Let $x$ be earnings - per - share and $y$ be dividends - per - share, and $n$ be the number of data points. Here $n = 6$. First, calculate the necessary sums: Let $x={1.00,4.08,3.45,8.29,1.6,2.9}$ and $y = {0.94,0.46,2.11,1.13,0.04,1.73}$. $\sum_{i = 1}^{6}x_i=1.00 + 4.08+3.45 + 8.29+1.6+2.9=21.32$. $\sum_{i = 1}^{6}y_i=0.94 + 0.46+2.11+1.13+0.04+1.73=6.41$. $\sum_{i = 1}^{6}x_i^{2}=1.00^{2}+4.08^{2}+3.45^{2}+8.29^{2}+1.6^{2}+2.9^{2}=1 + 16.6464+11.9025+68.7241+2.56+8.41=109.243$. $\sum_{i = 1}^{6}y_i^{2}=0.94^{2}+0.46^{2}+2.11^{2}+1.13^{2}+0.04^{2}+1.73^{2}=0.8836+0.2116+4.4521+1.2769+0.0016+2.9929=9.8187$. $\sum_{i = 1}^{6}x_iy_i=1.00\times0.94+4.08\times0.46+3.45\times2.11+8.29\times1.13+1.6\times0.04+2.9\times1.73=0.94 + 1.8768+7.2795+9.3677+0.064+5.017=24.545$.

Step2: Substitute into the formula

$n\sum_{i = 1}^{n}x_iy_i=6\times24.545 = 147.27$. $\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i=21.32\times6.41=136.6612$. $n\sum_{i = 1}^{n}x_i^{2}=6\times109.243 = 655.458$. $(\sum_{i = 1}^{n}x_i)^{2}=21.32^{2}=454.5424$. $n\sum_{i = 1}^{n}y_i^{2}=6\times9.8187 = 58.9122$. $(\sum_{i = 1}^{n}y_i)^{2}=6.41^{2}=41.0881$. $r=\frac{147.27-136.6612}{\sqrt{(655.458 - 454.5424)(58.9122 - 41.0881)}}$. $r=\frac{10.6088}{\sqrt{(200.9156)(17.8241)}}$. $r=\frac{10.6088}{\sqrt{3581.59}}$. $r=\frac{10.6088}{59.8464}\approx0.177$.

Answer:

$r\approx0.177$