Question

2. Linear trend regression Aa Aa The U.S. Census Bureau collects data on the size and location of the houses that are constructed each year in the United States. The table and corresponding plot that follow show annual time series data on the mean square feet of floor space in new one-family houses in the Midwest. 2,400 2,300 Floor Space (Mean Square Feet) 2,200 2,100 2,000 1996 1998 2000 2002 2004 2006 2008 Year applicable to the time series because the size of new one-family houses in the Smoothing methods Midwest has

Answer 1

year	t	yt	t-t	Y: -	(t-t (Y: -7	(t-t2
1996	1	2025	-6	-176.3	1057.8	36
1997	2	2065	-5	-136.3	681.5	25
1998	3	2125	-4	-76.3	305.2	16
1999	4	2135	-3	-66.3	198.9	9
2000	5	2170	-2	-31.3	62.6	4
2001	6	2209	-1	7.7	-7.7	1
2002	7	2209	0	7.7	0.0	0
2003	8	2198	1	-3.3	-3.3	1
2004	9	2222	2	20.7	41.4	4
2005	10	2310	3	108.7	326.1	9
2006	11	2290	4	88.7	354.8	16
2007	12	2328	5	126.7	633.5	25
2008	13	2331	6	129.7	778.2	36
Total	91	28617	0	0	4429	182
Mean	7	2201.3

Answer(a):

The formula for the linear trend line is as follows:

b, : Σt – t) - y) Σ(t −t)2

4429 bu 182

b1 = 24.3

Answer(b):

The value of intercept can be obtained as:

bo=y-bux

bo = 2201.3 - 24.3 * 7

bo = 2201.3 - 170.1

bo = 2031.2

Answer(c):

From the period 1996 to 2008, it is 13 years and slope is the value which indicates average increase per year, hence the average increase over the period = 13*24.3 = 315.9

Over the period from 1996 to 2008, the average square feet of floor space in new, one family houses in the Midwest increased by an average of 315.9.

Answer(d):

We have to forecast the mean square feet of new one family homes for 2009 using the estimated regression equation. That means we have to predict y_t when t=14

The regression equation is

Vt = 2031.2 + 24.3t

Vt = 2031.2 + 24.3 * 14

= 2371.4

The forecast is 2371.4 square feet.