To determine what percentage of the variability in $y$ (the response variable) can be explained by the linear relationship between $x$ (the predictor variable), we need to find the coefficient of determination, denoted as $R^2$. This is typically computed from the linear regression model for the given data.

Steps:

1. Find the linear regression equation from the data points $(x, y)$.
2. Compute $R^2$ from the linear regression model.
3. Convert $R^2$ to a percentage and round it to two decimal places.

Given the dataset:

x & : 8, 6, 10, 2, 6 \\ y & : 11, 15, 19, 10, 9 \end{aligned}$$ Let me calculate the regression line and $$R^2$$.Approximately **43.35%** of the proportion of variability in $$y$$ can be explained by the linear relationship between $$x$$ and $$y$$, based on this data set. Would you like further details on this calculation or any additional explanations? Here are five related questions to deepen understanding: 1. What is the meaning of the coefficient of determination ($$R^2$$) in regression analysis? 2. How does the value of $$R^2$$ relate to the strength of the linear relationship between variables? 3. What methods exist for improving the $$R^2$$ value in a regression model? 4. How does the interpretation of $$R^2$$ differ in nonlinear relationships? 5. Can an $$R^2$$ value ever be negative, and if so, what does that imply? **Tip:** A higher $$R^2$$ indicates a stronger linear fit, but it doesn't necessarily mean the model is good — always check the residuals and other factors!