Consider the following data X 4 6 8 10 12 15 20 25 39 48 Y 1

Consider the following data:

X 4 6 8 10 12 15 20 25 39 48

Y 12 21 25 34 40 48 65 75 100 160

Calculate:

(i) Least squares regression line by solving the normal equations

Show your work

Solution

Step 1: Find X?Y and X2 as it was done in the table below.

Step 2: Find the sum of every column:

?X=187 , ?Y=580 , ?X?Y=16669 , ?X2=5435

Step 3: Use the following equations to find a and b:

ab=?Y??X2??X??XYn??X2?(?X)2=580?5435?187?1666910?5435?1872?1.816=n??XY??X??Yn??X2?(?X)2=10?16669?187?58010?5435?(187)2?3.004

Step 4: Substitute a and b in regression equation formula

y = a + b?x= 1.816 + 3.004?x

Hence slope =3.004

and intercept = 1.816