Absolutely! Let’s break this down step by step. --- ### (a) **Interpreting the Model $Y = \beta_1 X$ (with $\beta_0 = 0$)** #### **Original Model** Normally, the simple linear regression model is: \[ Y = \beta_0 + \beta_1 X + \varepsilon \] where: - $\beta_0$ is the **intercept**, - $\beta_1$ is the **slope**, - $\varepsilon$ is the **error term**. #### **What if $\beta_0 = 0$?** Setting $\beta_0 = 0$ gives: \[ Y = \beta_1 X + \varepsilon \] Ignoring the error term for now (i.e., considering the average relationship), **the regression line passes through the origin (0, 0)**. That means: - When $X = 0$, the predicted value of $Y$ is also $0$. - The line is "forced" to go through the point (0, 0). #### **When is this appropriate?** This might be relevant if, for your data or context, you know that whenever $X$ is zero, $Y$ must also be zero due to physical, logical, or theoretical reasons. --- ### (b) **Finding the Least Squares Estimate for $\beta_1$ (with $\beta_0 = 0$)** #### **Objective** We want to find the value of $\beta_1$ that makes the predicted values as close as possible to the observed $Y$ values. We do this by **minimizing the sum of squared errors**: \[ \text{Minimize} \quad S(\beta_1) = \sum_{i=1}^n (Y_i - \beta_1 X_i)^2 \] #### **How do we find the minimum?** Take the derivative of $S(\beta_1)$ with respect to $\beta_1$, set it to zero, and solve for $\beta_1$. #### **Step-by-step:** 1. **Expand the sum:** \[ S(\beta_1) = \sum_{i=1}^n (Y_i - \beta_1 X_i)^2 \] 2. **Differentiate with respect to $\beta_1$:** \[ \frac{dS}{d\beta_1} = \sum_{i=1}^n 2 (Y_i - \beta_1 X_i)(-X_i) \] \[ = -2 \sum_{i=1}^n X_i (Y_i - \beta_1 X_i) \] 3. **Set the derivative to zero:** \[ -2 \sum_{i=1}^n X_i (Y_i - \beta_1 X_i) = 0 \] \[ \sum_{i=1}^n X_i Y_i - \beta_1 \sum_{i=1}^n X_i^2 = 0 \] 4. **Solve for $\beta_1$:** \[ \sum_{i=1}^n X_i Y_i = \beta_1 \sum_{i=1}^n X_i^2 \] \[ \boxed{ \hat{\beta}_1 = \frac{ \sum_{i=1}^n X_i Y_i }{ \sum_{i=1}^n X_i^2 } } \] --- ### **Summary** - **(a):** If $\beta_0 = 0$, the regression line must pass through (0,0). - **(b):** The least-squares estimate for $\beta_1$ is: \[ \hat{\beta}_1 = \frac{ \sum X_i Y_i }{ \sum X_i^2 } \] This formula gives you the slope when the intercept is restricted to zero. --- Let me know if you’d like to see a worked example or have any follow-up questions!