1. **Stating the problem:** We want to develop a simple regression model to predict a dependent variable $y$ based on an independent variable $x$. 2. **Formula used:** The simple linear regression model is given by: $$y = \beta_0 + \beta_1 x + \epsilon$$ where $\beta_0$ is the intercept, $\beta_1$ is the slope, and $\epsilon$ is the error term. 3. **Important rules:** - The relationship between $x$ and $y$ is assumed linear. - Errors $\epsilon$ are assumed to be normally distributed with mean zero. 4. **Steps to develop the model:** - Calculate the means $\bar{x}$ and $\bar{y}$. - Compute the slope: $$\beta_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$ - Compute the intercept: $$\beta_0 = \bar{y} - \beta_1 \bar{x}$$ 5. **Interpretation:** - $\beta_1$ tells how much $y$ changes for a unit change in $x$. - $\beta_0$ is the predicted value of $y$ when $x=0$. 6. **Final model:** $$y = \beta_0 + \beta_1 x$$ This model can be used to predict $y$ for new values of $x$.