6. ( 10 points) A student started to compute the correlation between variables \( x \) and \( y \). The work completed by the student is shown below. \[ \begin{array}{|c|c|} \hline z_{x} & z_{y} \\ \hline-1.8 & 0.5 \\ \hline-0.7 & -0.1 \\ \hline 0.1 & 0.6 \\ =0.07 \\ \hline 1.8 & -0.5 \\ \hline 2.2 & -0.1 \\ =0.06 \\ & =0.22 \\ & =0.9 \\ \text { sum }=-2.6 \end{array} \] a. Complete the computation of the correlation coefficient between variables. (4 decimal places.) \[ r=\frac{1}{5-1}(-1.45)=0.25(-1.45)=50.3625 \] b. Based on the correlation coefficient, classify the strength and the direction of the linear relationship. Strength \( = \) Weak Direction \( = \) negasi ve c. What percentage of the variation in \( y \) is explained by a linear relationship between \( x \) and \( y \) ? ( 4 significant figures.)

To solve the problem, we will follow the steps outlined in the prompt. ### Part a: Compute the Correlation Coefficient The correlation coefficient \( r \) is calculated using the formula: \[ r = \frac{1}{n-1} \sum (z_x \cdot z_y) \] where \( n \) is the number of pairs of data points, \( z_x \) and \( z_y \) are the standardized values of \( x \) and \( y \). From the table, we can see the values of \( z_x \) and \( z_y \): \[ \begin{array}{|c|c|} \hline z_{x} & z_{y} \\ \hline -1.8 & 0.5 \\ \hline -0.7 & -0.1 \\ \hline 0.1 & 0.6 \\ \hline 1.8 & -0.5 \\ \hline 2.2 & -0.1 \\ \hline \end{array} \] Now, we will calculate the products \( z_x \cdot z_y \): 1. \( -1.8 \cdot 0.5 = -0.9 \) 2. \( -0.7 \cdot -0.1 = 0.07 \) 3. \( 0.1 \cdot 0.6 = 0.06 \) 4. \( 1.8 \cdot -0.5 = -0.9 \) 5. \( 2.2 \cdot -0.1 = -0.22 \) Now, we sum these products: \[ \text{Sum} = -0.9 + 0.07 + 0.06 - 0.9 - 0.22 = -1.89 \] Now, we can calculate \( r \): \[ r = \frac{1}{5-1} \cdot (-1.89) = \frac{-1.89}{4} = -0.4725 \] ### Part b: Classify the Strength and Direction of the Linear Relationship The correlation coefficient \( r = -0.4725 \) indicates: - **Strength**: Weak (since \( |r| < 0.5 \)) - **Direction**: Negative (since \( r < 0 \)) ### Part c: Percentage of Variation Explained The percentage of variation in \( y \) explained by the linear relationship between \( x \) and \( y \) is given by \( r^2 \): \[ r^2 = (-0.4725)^2 = 0.2236 \] To express this as a percentage: \[ \text{Percentage} = r^2 \times 100 = 0.2236 \times 100 = 22.36\% \] ### Final Answers - **Part a**: \( r = -0.4725 \) - **Part b**: Strength = Weak, Direction = Negative - **Part c**: 22.36%