Question

Given the following five pairs of (x, y) values, x 1 3 11 8 14 y 10 7 4 2 1

(a)	Determine the least squares regression line.   (Be sure to save your unrounded values of b0 and b1 for use in Problem #6 below.)

(b)

Draw the least squares regression line accurately on a scatterplot. Then look to see which (x, y) pairs are above the regression line. Then add up the y-values for all of the (x, y) pairs that fall above the regression line. For example, if you draw your least squares regression line accurately on a scatterplot, and you find that the first two (x, y) pairs [i.e., (1, 10) and (3, 7)] are above the regression line, then since the sum of the two corresponding y-values is 10 + 7 = 17, you would enter 17 into the answer box.

(c) Calculate the residuals. (d) Calculate the residual sum of squares SS(error). (e) Find the value of the test statistic for testing the hypothesis H0 : ρ = 0 H1 : ρ ≠ 0 (f) Find the 10% critical value for the hypothesis test in (e)

Answer 1

(a)

(b)

Scatterplot is drawn as :

Here, we see that there 3 points which fall above the regression line : (1,10) , (11,4) and (14,1)

So, Sum of y values for the (x, y) pairs that fall above the regression line = 10 + 4 + 1 = 15

(c)

Predicted y = 9.384 - 0.6195 x

So, the residuals obtained are :

	x	y	Predicted y	Residual= (y - Pred y )
	1	10	8.765	1.235
	3	7	7.526	-0.526
	11	4	2.570	1.430
	8	2	4.428	-2.428
	14	1	0.712	0.288
Total	37	24	24	0

(d)

Residual= (y - Pred y )	(Residual)2
1.235	1.5264
-0.526	0.2763
1.430	2.0450
-2.428	5.8968
0.288	0.0832
0	9.8276

So, residual sum of squares SS(error) = 9.8276

(e)

First of all we find the value of r = Correlation coefficient

Also we have n = 5

Putting the above values in the formula :

We get r = - 0.906

Now, we conduct the test :

Test statistic is given as :

Critical value for 10 % level of significance =

Since, Test statistic | t | = | -3.705 | > 2.353 , we reject Ho and concldue that  

Hence, there is significant linear relationship between x and y.