Question

A consumer buying cooperative tested the effective heating area of 20 different electric space heaters with different wattages. Here are the results.

Heater	Wattage		Area
1	1,000		290
2	750		292
3	1,500		148
4	1,250		246
5	1,250		203
6	750		85
7	1,250		237
8	1,000		139
9	1,500		64
10	1,000		171
11	1,750		163
12	1,250		175
13	750		264
14	1,500		50
15	1,750		163
16	1,500		177
17	1,250		118
18	1,750		122
19	1,000		144
20	1,500		103

(a) Compute the correlation between the wattage and heating area. Is there a direct or an indirect relationship? (Negative values should be indicated by a minus sign. Round your answer to 3 decimal places.) The correlation of Wattage and Area is?

(b) Conduct a test of hypothesis to determine if it is reasonable that the coefficient is greater than zero. Use the 0.02 significance level. (Negative values should be indicated by a minus sign. Round your answer to 3 decimal places.)

(c) Develop the regression equation for effective heating based on wattage. (Negative values should be indicated by a minus sign. Round your answers to 3 decimal places.) The regression equation is?

(d) Which heater looks like the “best buy” based on the size of the residual? (Round residual value to 2 decimal places.) The ______heater is the "best buy." It heats an area that is________ square feet larger than estimated by the regression equation.

Answer 1

a)

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
1,000	290	68906.25	14957.29	-32103.75
750	292	262656.25	15450.49	-63703.75
1,500	148	56406.25	388.09	-4678.75
1,250	246	156.25	6130.89	-978.75
1,250	203	156.25	1246.09	-441.25
750	85	262656.25	6839.29	42383.75
1,250	237	156.25	4802.49	-866.25
1,000	139	68906.25	823.69	7533.75
1,500	64	56406.25	10753.69	-24628.75
1,000	171	68906.25	10.89	-866.25
1,750	163	237656.25	22.09	-2291.25
1,250	175	156.25	53.29	-91.25
750	264	262656.25	9273.69	-49353.75
1,500	50	56406.25	13853.29	-27953.75
1,750	163	237656.25	22.09	-2291.25
1,500	177	56406.25	86.49	2208.75
1,250	118	156.25	2470.09	621.25
1,750	122	237656.25	2088.49	-22278.75
1,000	144	68906.25	561.69	6221.25
1,500	103	56406.25	4186.09	-15366.25

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	25250	3354	2059375	94020.2	-188925.00
mean	1262.50	167.70	SSxx	SSyy	SSxy

correlation coefficient ,    r = Sxy/√(Sx.Sy) =   -0.429

there is indirect relationship

b)

Ho:   ρ = 0  
Ha:   ρ > 0  
n=   20  
alpha,α =    0.02  
correlation , r=   -0.429

t-test statistic = r*√(n-2)/√(1-r²) =        -2.017
DF=n-2 =   18  
p-value =    0.9706  
Decison:   P value > α, So, Do not reject Ho  
conclusion: it is not reasonable that the coefficient is greater than zero

c)

sample size ,   n =   20          
here, x̅ = Σx / n=   1262.50   ,     ȳ = Σy/n =   167.70  
                  
SSxx =    Σ(x-x̅)² =    2059375.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   -188925.0          
                  
estimated slope , ß1 = SSxy/SSxx =   -188925.0   /   2059375.000   =   -0.09174
                  
intercept,   ß0 = y̅-ß1* x̄ =   283.52049          
                  
so, regression line is   Ŷ =   283.520   +   -0.092 *x

d)

x	y				Ŷ	residual,ei=y-yhat
1,000	290				191.781	98.219
750	292				214.716	77.284
1,500	148				145.912	2.088
1,250	246				168.847	77.153
1,250	203				168.847	34.153
750	85				214.716	-129.716
1,250	237				168.847	68.153
1,000	139				191.781	-52.781
1,500	64				145.912	-81.912
1,000	171				191.78	-20.781
1,750	163				122.98	40.023
1,250	175				168.85	6.153
750	264				214.716	49.284
1,500	50				145.912	-95.912
1,750	163				122.977	40.023
1,500	177				145.912	31.088
1,250	118				168.847	-50.847
1,750	122				122.977	-0.977
1,000	144				191.781	-47.781
1,500	103				145.912	-42.912

The ___first___heater is the "best buy." It heats an area that is___98.22_____ square feet larger than estimated by the regression equation