Question

Calculate the residual for x=3 when given

Hours,x and Scores, y (3,60),(7,92),(6,88),(5,81),(6,85),(4,74),(5,82),(7,94),(4,76),(5,84)

Answer 1

Mean $\bar x=\dfrac{\sum x_i}{n}=\dfrac{3+7+6+5+6+4+5+7+4+5}{10}=\dfrac{52}{10}=5.2$

Mean $\bar y =\dfrac{\sum y_i}{n}=\dfrac{816}{10}=81.6$

$\bar x=5.2$ is not an integer , use assumed mean A=5

$\bar y = 81.6$ is not an integer , use assumed mean B = 82

Now,

From table:

$b_{yx}=\dfrac{n\sum dxdy-(\sum dx)(\sum dy)}{n\sum dx^2-(\sum dx)^2}\\\ \\=\dfrac{(10\times 111)-2\times (-4)}{(10\times 16)-(2)^2}\\\ \\=\dfrac{1110+8}{160-4}\\\ \\=\dfrac{1118}{156}=7.1667$

Regression line y on x :

$y-\bar y =b_{yx}(x-\bar x)\\\Rightarrow y-81.6=7.1667(x-5.2)\\\Rightarrow \boxed{ y=7.1667x+44.3333}$

Now estimate y for x=3

$y=7.1667\times 3+44.3333\\\Rightarrow y=21.5+44.3333\\\Rightarrow \boxed{y=65.8333}$

So, the residual at x=3

$\epsilon_0=|y-\hat y|$

From data when x=3 y= 60

And residual is the absolute value of a residual measures the vertical distance between the actual value of y and the estimated value of y

So residual at x=3

$\Rightarrow |65.833-60|=5.833$