Question

David's Landscaping has collected data on home values (in thousands of $) and expenditures (in thousands of $) on landscaping with the hope of developing a predictive model to help marketing to potential new clients. Data for 14 households may be found in the file Landscape.

Home Value ($1,000)	Landscaping Expenditures ($1,000)
242	8.1
321	10.8
198	12.2
340	16.2
300	15.6
400	18.9
800	23.5
200	9.5
521	17.5
547	22.0
437	12.1
464	13.5
635	17.9
356	13.9

(a)

Develop a scatter diagram with home value as the independent variable.

A scatter diagram has a horizontal axis labeled "Landscaping Expenditures ($1,000)" with values from 0 to 25 and a vertical axis labeled "Home Value ($1,000)" with values from 0 to 900. The scatter diagram has 14 points. A pattern goes up and right from (8.1, 198) to (23.5, 800). The points are scattered moderately from the pattern.

A scatter diagram has a horizontal axis labeled "Landscaping Expenditures ($1,000)" with values from 0 to 25 and a vertical axis labeled "Home Value ($1,000)" with values from 0 to 900. The scatter diagram has 14 points. A pattern goes down and right from (6.5, 800) to (21.9, 198). The points are scattered moderately from the pattern.

A scatter diagram has a horizontal axis labeled "Home Value ($1,000)" with values from 0 to 900 and a vertical axis labeled "Landscaping Expenditures ($1,000)" with values from 0 to 25. The scatter diagram has 14 points. A pattern goes up and right from (198, 8.1) to (800, 23.5). The points are scattered moderately from the pattern.

A scatter diagram has a horizontal axis labeled "Home Value ($1,000)" with values from 0 to 900 and a vertical axis labeled "Landscaping Expenditures ($1,000)" with values from 0 to 25. The scatter diagram has 14 points. A pattern goes down and right from (198, 21.9) to (800, 6.5). The points are scattered moderately from the pattern.

(b)

What does the scatter plot developed in part (a) indicate about the relationship between the two variables?

The scatter diagram indicates a negative linear relationship between home value and landscaping expenditures.The scatter diagram indicates a positive linear relationship between home value and landscaping expenditures.    The scatter diagram indicates a nonlinear relationship between home value and landscaping expenditures.The scatter diagram indicates no apparent relationship between home value and landscaping expenditures.

(c)

Use the least squares method to develop the estimated regression equation. (Let x = home value (in thousands of $), and let y = landscaping expenditures (in thousands of $). Round your numerical values to five decimal places.)

ŷ =

  

(d)

For every additional $1,000 in home value, estimate how much additional will be spent (in $) on landscaping. (Round your answer to the nearest cent.)

$  

(e)

Use the equation estimated in part (c) to predict the landscaping expenditures (in $) for a home valued at $575,000. (Round your answer to the nearest dollar.)

$

Answer 1

A.scatter diagram with home value as the independent variable.

Scatter diagram has a horizontal axis labeled "Landscaping Expenditures ($1,000)" with values from 0 to 25 and a vertical axis labeled "Home Value ($1,000)" with values from 0 to 900. The scatter diagram has 14 points. A pattern goes up and right from (8.1, 198) to (23.5, 800). The points are scattered moderately from the pattern.

A scatter diagram has a horizontal axis labeled "Home Value ($1,000)" with values from 0 to 900 and a vertical axis labeled "Landscaping Expenditures ($1,000)" with values from 0 to 25. The scatter diagram has 14 points. A pattern goes up and right from (198, 8.1) to (800, 23.5). The points are scattered moderately from the pattern.

B. What does the scatter plot developed in part (a) indicate about the relationship between the two variables?

The scatter diagram indicates no apparent relationship between home value and landscaping expenditures.

C.

Estimated regression equation ŷ can be determined as below

	HOME VALUE X	LANDSCAPING EXPENDITURE Y	X-X0	Y-Y0	(X-X0)2	(Y-Y0)2	(X-X0)*(Y-Y0)
	242	8.1	-169.5	-7.021429	28730.25	49.30046	1190.132143
	321	10.8	-90.5	-4.321429	8190.25	18.67474	391.0892857
	198	12.2	-213.5	-2.921429	45582.25	8.534745	623.725
	340	16.2	-71.5	1.078571	5112.25	1.163316	-77.11785714
	300	15.6	-111.5	0.478571	12432.25	0.229031	-53.36071429
	400	18.9	-11.5	3.778571	132.25	14.2776	-43.45357143
	800	23.5	388.5	8.378571	150932.3	70.20046	3255.075
	200	9.5	-211.5	-5.621429	44732.25	31.60046	1188.932143
	521	17.5	109.5	2.378571	11990.25	5.657602	260.4535714
	547	22	135.5	6.878571	18360.25	47.31474	932.0464286
	437	12.1	25.5	-3.021429	650.25	9.129031	-77.04642857
	464	13.5	52.5	-1.621429	2756.25	2.629031	-85.125
	635	17.9	223.5	2.778571	49952.25	7.720459	621.0107143
	356	13.9	-55.5	-1.221429	3080.25	1.491888	67.78928571
Mean	411.5	15.12143	SUM	382633.5	267.9236	8194.15

ŷ= b0+b1 X

b1=

b1= 8194.15/382633.5

b1=0.02142

Consider ŷ =15.12143, x = 411.5

Subsitute in ŷ= b0+b1 X

Now 15.12143= b0 + 0.02142 X 411.5

b0= 6.3071

ŷ= 6.3071+0.02142 X

D. For every additional $1,000 in home value, the landscapping expenditure will be-

ŷ= 6.3071+0.02142 X

=6.3071+0.02142*1000

=27.7271

=$27.73 (Rounded up to nearest cent)

E. For a home valued at $575,000

Landscapping Expenditure will be given by,

ŷ= 6.3071+0.02142 X

=6.3071+0.02142*575000

=12322.8071

=$12323 rounded up to nearest dollar