Question

Consider the relationship between the percentage of seats held in Parliament by the governing party (Y = % Seats) and their respective national popular vote (X = % NPV). Data from 7 Canadian federal elections (1993-2011) are provided in the table below. Assume that the error terms are approximately normally distributed with equal variance for all national popular vote values. Given the information, conduct a simple linear regression and estimate the coefficient of determination of the regression and explain what it indicates.

Percent of seats and of national popular vote for governing party (1993-2011)

Year 1993 1997 2000 2004 2006 2008 2011

(x) % NPV 41.24 38.46 40.85 36.73 36.27 37.65 39.62

(y) % Seats 60.00 51.50 57.14 43.83 40.26 46.43 53.90

Answer 1

First we will find simple linear regression

Sum of X = 270.82
Sum of Y = 353.06
Mean X = 38.6886
Mean Y = 50.4371
Sum of squares (SSX) = 22.8655
Sum of products (SP) = 83.5856

Regression Equation = ŷ = bX + a

b = SP/SSX = 83.59/22.87 = 3.6555

a = MY - bMX = 50.44 - (3.66*38.69) = -90.9903

ŷ = 3.6555X - 90.9903

Now to find coefficient of determination, we will first find correlation coefficient

X Values
∑ = 270.82
Mean = 38.689
∑(X - Mx)2 = SSx = 22.865

Y Values
∑ = 353.06
Mean = 50.437
∑(Y - My)2 = SSy = 312.783

X and Y Combined
N = 7
∑(X - Mx)(Y - My) = 83.586

R Calculation
r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))

r = 83.586 / √((22.865)(312.783)) = 0.9884

So coefficient of determination is r^2=0.9884^2=0.9769

Hence 97.69% of variation in y is explained by x.