Question

Distance	3.4	1.8	4.6	2.3	3.1	5.5	0.7	3.0
Damage	26.2	17.8	31.3	23.1	27.5	36.0	14.1	22.3

Distance	2.6	4.3	2.1	1.1	6.1	4.8	3.8
Damage	19.6	31.3	24.0	17.3	43.2	36.4	26.1

Write the equation (formula) for a residual and then calculate its value “by hand” for the observation in the data set whose distance between the fire and nearest fire station (in miles) is 3.0; show your work. Based on this value, was the observation overestimated (below average) or underestimated (above average) by the regression? Explain. See page 191 in the course text.

Answer 1

residual(e)=y-y^=y-(a+bx)=y-(10.26+4.92*x)

for x=3, the estimated value of y is y^=10.26+4.92*3=25.02

residual for this value is y-y^=22.3-25.02=-2.72, negative residual means it is overestimated

Residual = Observed – Predicted

positive values for the residual mean the prediction was too low, and negative values mean the prediction was too high.when the model overestimates the observation: the residual is negative.

here Damage(y) is dependent variable and Distance(x) is independent variable.

the regression equation y^=a+bx=10.26+4.92*x

a=(sum(y)*sum(x²)-sum(x)*sum(xy))/(n*sum(x²)-(sum(x))²)=(396.16*196.16-49.2*1470.65)/(15*196.16-49.2*49.2)=10.26

b=(n*sum(xy)-sum(x)*sum(y))/ (n*sum(x²)-(sum(x))²)=(15*1470.65-49.2*396.2)/(15*196.16-49.2*49.2)=4.92

S.N.	Distance(x)	Damage(y)	x2	y2	xy
1	3.4	26.2	11.56	686.44	89.08
2	1.8	17.8	3.24	316.84	32.04
3	4.6	31.3	21.16	979.69	143.98
4	2.3	23.1	5.29	533.61	53.13
5	3.1	27.5	9.61	756.25	85.25
6	5.5	36	30.25	1296	198
7	0.7	14.1	0.49	198.81	9.87
8	3.0	22.3	9	497.29	66.9
9	2.6	19.6	6.76	384.16	50.96
10	4.3	31.3	18.49	979.69	134.59
11	2.1	24	4.41	576	50.4
12	1.1	17.3	1.21	299.29	19.03
13	6.1	43.2	37.21	1866.24	263.52
14	4.8	36.4	23.04	1324.96	174.72
15	3.8	26.1	14.44	681.21	99.18
sum=	49.2	396.2	196.16	11376.5	1470.65
n=	15	15	15	15	15

a=(sum(y)*sum(x²)-sum(x)*sum(xy))/(n*sum(x²)-(sum(x))²)=(396.16*196.16-49.2*1470.65)/(15*196.16-49.2*49.2)=

b=(n*sum(xy)-sum(x)*sum(y))/ (n*sum(x²)-(sum(x))²)=(15*1470.65-49.2*396.2)/(15*196.16-49.2*49.2)=