a. To find the least squares regression equation for a group of (x,y) points:
Step 1: For each point(x,y), we need to calculate x² and xy first.
Step 2: Sum all x, y, x² and xy, which gives us Σx, Σy, Σx² and Σxy
Step 3: Calculate Slope m: m = (NΣxy − Σx Σy)/ N(Σx²) − (Σx)² , here N is the number of data.
Step 4: Calculate Intercept b: b = Σy − m(Σx)/ N
Step 5: Assemble the equation as y= mx+b
Here, we have ∑x = 120, ∑y= 1379, ∑x²= 919.5, ∑xy= 10466 and N= 16
So, m= (NΣxy − Σx Σy)/ N(Σx²) − (Σx)²
= [tex] \frac{(16*10466)-(120*1379)}{(16*919.5)-(120)^{2}} [/tex]
= [tex] \frac{167456-165480}{14712-14400} [/tex]
= [tex] \frac{1976}{312} [/tex]
= 6.33
and b = Σy − m(Σx)/ N
= [tex] \frac{1379-(6.33*120)}{16} [/tex]
= [tex] \frac{1379-759.6}{16} [/tex]
= [tex] \frac{619.4}{16} [/tex]
= 38.71
So, the least squares regression equation is : y= mx+b ⇒ y= 6.33x +38.71
b. When x= 8, then y= (6.33×8)+38.71 = 50.64 + 38.71 =89.35
The approximate test score of a student who sleeps an average of 8 h a night is 89.35%
