Question

8. If the data set shown below were used to fit the following simple regression model, y = Bo + Bix te, which of the following equations would result in the smallest SSE? a) y=2x V X b) y=2 + x 3 2 c) y = 3 + 0.5 4 2 d) All of the above 52

Answer 1

Solution:

The correct answer is d) All of the above

Explanation:

Let's first consider the equation:

$y=2x$

The predicted values of y are:

$\hat{y}_{1}=2 \times 2=4$

$\hat{y}_{2}=2 \times 2=4$

$\hat{y}_{3}=2 \times 2=4$

$\therefore SSE=\sum (y_{i}-\hat{y}_{i})^{2}$

  $=(3-4)^{2}+(4-4)^{2}+(5-4)^{2}$

= 1+0+1

$=2$

Now let's consider the second option:

y = 2+

The predicted values of y are:

Ý1 = 2+2 = 4

$\hat{y}_{2}=2 + 2=4$

$\hat{y}_{3}=2 + 2=4$

$\therefore SSE=\sum (y_{i}-\hat{y}_{i})^{2}$

  $=(3-4)^{2}+(4-4)^{2}+(5-4)^{2}$

= 1+0+1

$=2$

Now let's consider the third option:

$y=3+0.5x$

The predicted values of y are:

$\hat{y}_{1}=3 +0.5 \times 2=4$

$\hat{y}_{2}=3+ 0.5 \times 2=4$

$\hat{y}_{3}=3 +0.5 \times 2=4$

$\therefore SSE=\sum (y_{i}-\hat{y}_{i})^{2}$

  $=(3-4)^{2}+(4-4)^{2}+(5-4)^{2}$

= 1+0+1

$=2$

Since all the equations have same SSE, therefore, the option d) All of the above is correct