Question

The following table shows the average price of a two-bedroom apartment in a downtown area from 1994 to 2004 (t = 0 represents 1994). Yeart 0 2 4 6 10 Price P 0.98 1.00 1.20 1.55 1.80 2.20 ($ million) (a) Use technology to obtain the linear regression line and correlation coefficient r, with all coefficients rounded to two decimal places. p(t) = Plot the regression line and the given points. 6 8 10 6 8 10

Answer 1

Using the following regression formula:

i=1 (Ii - )(Yi - y) Li-(I; - 72

Year (X)	Price (Y)	(X-Xm)	Y-Ym	(X-Xm)(Y-Ym)	(X-Xm)^2	(Y-Ym)^2
0	0.98	-5	-0.475	2.375	25	0.225625
2	1	-3	-0.455	1.365	9	0.207025
4	1.2	-1	-0.255	0.255	1	0.065025
6	1.55	1	0.095	0.095	1	0.009025
8	1.8	3	0.345	1.035	9	0.119025
10	2.2	5	0.745	3.725	25	0.555025

Xm = Mean of X = 5 ;

Ym = Mean of Y = 1.455

Using the formula mentioned, we get

B= 0.1264

To compute the constant term, following formula is used

â=y-BT,

Thus, we get

$\hat{\alpha } = 0.822$

Thus, the regression equation is:

To compute the correlation coefficient, following formula is used

r= (x-)(Y-Y) (x-7)2 /(x-7)2 Where, X-mean of X variable Y mean of Y variable

We get r = 0.7868 ~ 0.79

The graph of the regression equation is :

р 3 2 4 6 8 10 N -11

This can be seen using the following table, where the value of 't' are put in the regression equation

Year (X)	Price (Y)	Equation
0	0.98	0.822
2	1	1.074
4	1.2	1.326
6	1.55	1.578
8	1.8	1.83
10	2.2	2.082

To find out the residual, we move a step forward, by subtracting the 'equation' value from the 'Price' value

Year (X)	Price (Y)	Equation	Residual
0	0.98	0.822	0.158
2	1	1.074	-0.074
4	1.2	1.326	-0.126
6	1.55	1.578	-0.028
8	1.8	1.83	-0.03
10	2.2	2.082	0.118

Thus, the answer to last part of the question is option a. The residual is positive at first, then turns negative and finally turns positive at last.

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