Question

For questions 8 - 15, use the following set of data to find the equation of the line of

best fit and assess the fit of the equation by plotting and analyzing residuals.
The data below show the mean swimming speed (m/s) of basking sharks of various
lengths (m)
Length (m) Speed (m/s) Predicted speed Residual
2.5
3
3.5
4.5
5
4.8
5.5
5.2
6
5.5
a Calculate the line of best fit Round a and b to the nearest hundredth.

Answer 1

Y^= 1.767 + 0.6294X when rounded give s Y^= 1.77 +0.63X

b= 0.6294 rounded to 0.63

a= 1.77

The predicted lines are for each X and Y

3.340,3.96, 4.914, 5.228, and 5.543

Explanation:

The data given is

Length (m) Speed (m/s) Predicted Line

2.5 3 3.340

3.5 4.5 3.96

5 4.8 4.914

5.5 5.2 5.228

6 5.5 5.543

The calculations are

Xsquare XY Y X

6.25 7.5 3 2.5

12.25 15.75 4.5 3.5

25 24 4.8 5

30.25 28.6 5.2 5.5

36 33 5.5 6

Total 109.75 108.85 23 22.5

The estimated regression line of Y on X is

Y^ = a +bX

and two normal equations are

∑Y = na + b∑X

∑XY= a∑X + b∑X²

Now X`= ∑X/ n= 22.5/5=4.5

Y`= ∑Y/ n= 23/5= 4.6

b= n∑XY- (∑X)(∑Y) / n∑X²- (∑X²)

Putting the values

b= 5(108.85) - (23)(22.5)/ 5(109.75)- (22.5)²

b= 544.25-517.5/ 548.75-506.25

b= 26.75 /42.5

b= 0.6294

and

a= Y`- bX~= 4.6- 0.6294(4.5)= 4.6-2.823= 1.767

Hence the

desired estimated regression line of Y on X is

Y^= 1.767 + 0.6294X

Y^= 1.77 +0.63X

The estimated regression co efficient b= 0.6294 indicates that the values of Y increase by 0.6294 units for a unit increase in X.