Ols, Intercept and Slope Estimate

Course: Quantitative Techniques

Assignment No. 1

Q.1 The following table contains the ACT scores and the GPA (grade point average) for eight college students. Grade point average is based on a four-point scale and has been rounded to one digit after the decimal. Student 1 2 3 4 5 6 7 GPA 2.8 3.4 3.0 3.5 3.6 3.0 2.7 ACT 21 24 26 27 29 25 25

(i)

8 3.7 30 _____________________________________________ Estimate the relationship between GPA and ACT using OLS; that is, obtain the intercept and slope estimates in the equation . Comment on the direction of the relationship. Does the intercept have a useful interpretation here? Explain. How much higher is the GPA predicted to be if the ACT score is increased by five points? Solution-(i) ACT = x and GPA = y Where Sample Size n=8 (given)

Let us suppose

In order to find;

ˆ ˆ ˆ GPA   0  1 ACT Or

ˆ ˆ =  0 + 1 x ------------Eq (1)

We proceed as following

x 21 24 26 27 29 25 25 30 =207 y 2.8 3.4 3 3.5 3.6 3 2.7 3.7 =25.7 x² 441 576 676 729 841 625 625 900 =5413 xy 58.8 81.6 78 94.5 104.4 75 67.5 111 =670.8

Course: Quantitative Techniques Now = = = = = 3.213 = 25.875

Course Code: MGT 605-QT

Assignment No. 1

Sample Slope i.e. =

ˆ 1 =

=

ˆ 1

= .1022

Now we need to calculate Sample Y – Intercept i.e.

ˆ 0

Now putting the values of ,

= – 1 , whereas:

ˆ and 1 in sample y-intercept formula we get,

ˆ

ˆ  0 = (3.213)-(0.1022)(25.875)

ˆ  0 =0.568

Putting values in Eq. (1) we get

GPA = .5681 + .1022 ACT

The intercept does not have a useful interpretation because ACT is not close to zero for the population of interest. If ACT is 5 points higher, GPA increases by .1022(5) = .511. (ii) Compute the fitted values and residuals for each observation, and verify that the residuals (approximately) sum to zero. Solution-(ii) y 2.8 3.4 3 3.5 3.6 3 2.7 3.7

ˆ ˆ =  0 + 1 x (0.568)+(0.1022)(21)=2.714 (0.568)+(0.1022)(24)=3.020 (0.568)+(0.1022)(26)=3.225...