The quantity $ \varepsilon$
in the simple linear regression model $Y = \beta _ { 0 } + \beta _ { 1 } x + \varepsilon$ is a random variable, assumed to be normally distributed with $E ( \varepsilon ) = 0 \text { and } V ( \varepsilon ) = \sigma ^ { 2 }$ The estimated standard deviation $\hat { \sigma }$ is given by
A) SSE / (n - 2)
B) $\sqrt { S S E / ( n - 2 ) }$
C) $[ S S E / ( n - 2 ) ] ^ { 2 }$
D) $\operatorname { SSE } / \sqrt { n - 2 }$
E) $\sqrt { \operatorname { SSE } } / ( n - 2 )$

Question

The quantity $ \varepsilon$  
 in the simple linear regression model $Y = \beta _ { 0 } + \beta _ { 1 } x + \varepsilon$ is a random variable, assumed to be normally distributed with $E ( \varepsilon ) = 0 \text { and } V ( \varepsilon ) = \sigma ^ { 2 }$ The estimated standard deviation $\hat { \sigma }$ is given by
A) SSE / (n - 2)
B) $\sqrt { S S E / ( n - 2 ) }$
C) $[ S S E / ( n - 2 ) ] ^ { 2 }$
D) $\operatorname { SSE } / \sqrt { n - 2 }$
E) $\sqrt { \operatorname { SSE } } / ( n - 2 )$

Quizplus · Accepted Answer

The estimated standard deviation of the errors, $\hat{\sigma}$, in a simple linear regression model is calculated as the square root of the sum of squared errors (SSE) divided by the degrees of freedom, which is $n - 2$ for simple linear regression. Therefore, $\hat{\sigma} = \sqrt{SSE / (n - 2)}$.

The Quantity ε \varepsilonε In the Simple Linear Regression Model Y=β0+β1x+εY = \beta _ { 0 } + \beta _ { 1 } x + \varepsilonY=β0​+β1​x+ε

The Quantity $\varepsilon$ In the Simple Linear Regression Model $Y = \beta _ { 0 } + \beta _ { 1 } x + \varepsilon$