Let's tackle each part of the question step by step: --- **(a)** *Assume that \( X = 0 \) is within the scope of the model. What is the implication for the regression function if \( \beta_0 = 0 \)?* **Answer:** If \( \beta_0 = 0 \), the regression equation becomes: \[ Y_i = \beta_1 X_i + \epsilon_i \] When \( X = 0 \), \( Y_i = \epsilon_i \). Thus, the regression line passes through the origin (0,0) on the graph (assuming \( \epsilon_i \) has mean zero). This means there is no intercept term; all predictions are forced through the origin. --- **(b)** *Under the assumption of \( \beta_0 = 0 \), derive the least squares estimate of \( \beta_1 \).* **Answer:** When \( \beta_0 = 0 \), the model is \( Y_i = \beta_1 X_i + \epsilon_i \). The least squares estimate minimizes: \[ \sum_{i=1}^n (Y_i - \beta_1 X_i)^2 \] Set the derivative with respect to \( \beta_1 \) to zero: \[ -2 \sum_{i=1}^n X_i (Y_i - \beta_1 X_i) = 0 \implies \sum_{i=1}^n X_i Y_i = \beta_1 \sum_{i=1}^n X_i^2 \implies \hat{\beta}_1 = \frac{\sum_{i=1}^n X_i Y_i}{\sum_{i=1}^n X_i^2} \] --- **(c)** *Find its mean and variance. Is it unbiased?* **Answer:** Recall \( Y_i = \beta_1 X_i + \epsilon_i \) and \( \epsilon_i \) has mean 0, variance \( \sigma^2 \). \[ \hat{\beta}_1 = \frac{\sum_{i=1}^n X_i Y_i}{\sum_{i=1}^n X_i^2} = \frac{\sum_{i=1}^n X_i (\beta_1 X_i + \epsilon_i)}{\sum_{i=1}^n X_i^2} = \beta_1 + \frac{\sum_{i=1}^n X_i \epsilon_i}{\sum_{i=1}^n X_i^2} \] - Expectation: \[ E[\hat{\beta}_1] = \beta_1 + \frac{E[\sum X_i \epsilon_i]}{\sum X_i^2} = \beta_1 \] so it **is unbiased**. - Variance: \[ Var(\hat{\beta}_1) = Var\left( \frac{\sum X_i \epsilon_i}{\sum X_i^2} \right) = \frac{\sum X_i^2 Var(\epsilon_i)}{(\sum X_i^2)^2} = \frac{\sigma^2}{\sum X_i^2} \] --- **(d)** *How do you fit such a model using the lm function?* **Answer:** In R, you can fit a regression through the origin (no intercept) via: ```R lm(Y ~ X - 1) ``` or equivalently: ```R lm(Y ~ 0 + X) ``` --- **(e)** *What is the implication for the regression function if \( \beta_1 = 0 \)? How would the regression function plot on a graph?* **Answer:** If \( \beta_1 = 0 \), the model is \( Y_i = \beta_0 + \epsilon_i \). The regression function is a **horizontal line** at \( Y = \beta_0 \). This means \( Y \) does not depend on \( X \) at all. --- **(f)** *Under the assumption of \( \beta_1 = 0 \), derive the least squares estimate of \( \beta_0 \).* **Answer:** With \( Y_i = \beta_0 + \epsilon_i \) (no \( X \) term), minimize: \[ \sum (Y_i - \beta_0)^2 \] Set derivative to zero: \[ -2\sum (Y_i - \beta_0) = 0 \implies \sum Y_i = n \beta_0 \implies \hat{\beta}_0 = \frac{1}{n} \sum Y_i = \bar{Y} \] --- **(g)** *Find the Expectation and Variance of the estimate in (f). Is it unbiased?* **Answer:** \[ \hat{\beta}_0 = \bar{Y} \] - Expectation: \[ E[\hat{\beta}_0] = E[\bar{Y}] = \beta_0 + E[\bar{\epsilon}] = \beta_0 \text{ (since } E[\bar{\epsilon}] = 0) \] So it is **unbiased**. - Variance: \[ Var(\hat{\beta}_0) = Var(\bar{Y}) = Var\left( \frac{1}{n} \sum \epsilon_i \right) = \frac{\sigma^2}{n} \] --- Let me know if you need any step-by-step calculation!