Question

1. Suppose we have the following data on variable X (independent) and variable Y (dependent):

X	Y
2	70
0	70
4	130

a.) By hand, determine the simple regression equation relating Y and X.

b.) Calculate the R-Square measure and interpret the result.

c.) Calculate the adjusted R-Square.

d.) Test to see whether X and Y are significantly related using a test on the population correlation. Test this at the 0.05 level.

e.) Test to see whether X and Y are significantly related using a t-test on the slope of X. Test this at the 0.05 level.

f.) Test to see whether X and Y are significantly related using an F-Test on the slope of X. Test this at the 0.05 level.

Answer 1

a) The least square estimates for a line Y on x - Y=a+bx are

	2	70	0	400	0	90	0
	0	70	4	400	40	60	900
	4	130	4	1600	80	120	900
Total	6	270	8	2400	120		1800
average	2	90

So regression line for this sample : Y= 60 + 15x

b) Coefficient of determination:

interpretation :

75% of the variation in y is 'explained by' the variation in predictor x.

c)  

where, N= number of observations=3.

p= number of predictors = 1

d)

test statistic to test this hypothesis:

, under H₀

From sample, n=3, r= 0.866, T= 1.7318

P-value= 2*min(P[T>1.7318],P[T<1.7318])=2*min(0.1667,0.8333)=0.3334 > 0.05

So at 5% level of significance, we can say that X and Y not very correlated since we fail to reject the null hypothesis.