Find the value of the linear correlation coefficient \( r \) using the paired data below. x 75359516356516 y 165146136177195157115 0.109 -0.078 -0.086 0.214

We are given the following paired data (by splitting the digits into numbers): - \( x \): 75, 35, 95, 163, 56, 5, 16 - \( y \): 165, 146, 136, 177, 195, 157, 115 We wish to compute the linear correlation coefficient \[ r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{\Bigl[n\sum x^2 - (\sum x)^2\Bigr]\Bigl[n\sum y^2 - (\sum y)^2\Bigr]}}. \] Below are the step‐by‐step calculations. --- ### 1. Compute the sums There are \( n = 7 \) pairs. **Sum of \( x \):** \[ \sum x = 75 + 35 + 95 + 163 + 56 + 5 + 16. \] Step by step: \[ 75 + 35 = 110, \quad 110 + 95 = 205, \quad 205 + 163 = 368, \quad 368 + 56 = 424, \quad 424 + 5 = 429, \quad 429 + 16 = 445. \] Thus, \[ \sum x = 445. \] **Sum of \( y \):** \[ \sum y = 165 + 146 + 136 + 177 + 195 + 157 + 115. \] Step by step: \[ 165 + 146 = 311, \quad 311 + 136 = 447, \quad 447 + 177 = 624, \quad 624 + 195 = 819, \quad 819 + 157 = 976, \quad 976 + 115 = 1091. \] Thus, \[ \sum y = 1091. \] --- ### 2. Compute the sum of squares **Sum of squares for \( x \):** \[ \begin{array}{rcl} 75^2 &=& 5625,\\[1mm] 35^2 &=& 1225,\\[1mm] 95^2 &=& 9025,\\[1mm] 163^2 &=& 26569,\\[1mm] 56^2 &=& 3136,\\[1mm] 5^2 &=& 25,\\[1mm] 16^2 &=& 256. \end{array} \] Then, \[ \sum x^2 = 5625 + 1225