1. **Problem statement:** We have a moderate positive linear correlation between variables $x$ and $y$ with regression line $$y = 2x + 1$$ determined from data where $$3 \leq x \leq 12$$. 2. **Formula and rules:** The regression line equation $$y = mx + b$$ models the relationship between $x$ and $y$ within the observed data range. It is appropriate to use it for prediction only within this range (interpolation). 3. **(a) Why is it appropriate to find the regression line?** - Because there is a moderate positive linear correlation, a linear model fits the data well. - The regression line summarizes the trend of $y$ changing with $x$. - It helps predict $y$ values for $x$ within the observed range $3 \leq x \leq 12$. 4. **(b) Why is it not reliable to estimate $y$ when $x=0$?** - $x=0$ is outside the observed data range (extrapolation). - The linear relationship may not hold outside $3 \leq x \leq 12$. - Predictions at $x=0$ may be inaccurate or misleading. 5. **(c) Why is it reliable to estimate $y$ when $x=10$?** - $x=10$ lies within the observed data range. - The regression line is based on data including $x=10$. - Predictions here are interpolation and generally reliable. 6. **(d) Why is it not valid to estimate $x$ when $y=20$?** - The regression line models $y$ as a function of $x$, not $x$ as a function of $y$. - Estimating $x$ from $y$ requires the inverse regression or another model. - Using this regression line to find $x$ from $y$ ignores error structure and can be invalid. **Final answers:** - (a) Appropriate because of moderate positive linear correlation and data range. - (b) Not reliable for $x=0$ as it is outside data range (extrapolation). - (c) Reliable for $x=10$ as it is within data range (interpolation). - (d) Not valid to estimate $x$ from $y$ using this regression line.