Question

(a) Given a simple linear regression model $y = \beta_0 + \beta_1x + \varepsilon$, where the intercept $\beta_0$ and the slope $\beta_1$ are unknown constants and $\varepsilon$ is a random error component. Show that: (i) $Var(\hat{\beta_1}) = \frac{\sigma^2}{\sum_{i=1}^{n}(x_i - \bar{x})^2}$ (ii) $Var(\hat{\beta_0}) = \sigma^2 \left[ \frac{1}{n} + \frac{\bar{x}^2}{\sum_{i=1}^{n}(x_i - \bar{x})^2} \right]$

          (a) Given a simple linear regression model $y = \beta_0 + \beta_1x + \varepsilon$, where the intercept $\beta_0$ and the slope $\beta_1$ are unknown constants and $\varepsilon$ is a random error component. Show that:
(i) $Var(\hat{\beta_1}) = \frac{\sigma^2}{\sum_{i=1}^{n}(x_i - \bar{x})^2}$
(ii) $Var(\hat{\beta_0}) = \sigma^2 \left[ \frac{1}{n} + \frac{\bar{x}^2}{\sum_{i=1}^{n}(x_i - \bar{x})^2} \right]$

(a) Given a simple linear regression model y = β0 + β1x + ε, where the intercept β0 and the slope β1 are unknown constants and ε is a random error component. Show that:
(i) Var(β̂1̂) = (σ^2)/(∑i=1^n(xi - x̅)^2)
(ii) Var(β̂0̂) = σ^2 [ (1)/(n) + (x̅^2)/(∑i=1^n(xi - x̅)^2)]

Added by Cristina M.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

09/04/2023

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Given a simple linear regression model y = Î²0 + Î²1x + Îµ, where the intercept Î²0 and the slope Î²1 are unknown constants and Îµ is a random error component. Show that: (i) (5 marks) (ii) Var(5 marks)