Question

The efficiency for a steel specimen immersed in a phosphating tank is the weight of the phosphate coating divided by the metal loss (both in mg/ft2). An article gave the accompanying data on tank temperature (x) and efficiency ratio (y).

Temp.	174	176	177	178	178	179	180	181
Ratio	0.84	1.33	1.52	1.11	1.05	1.14	1.10	1.74

Temp.	184	184	184	184	184	185	185	186
Ratio	1.41	1.50	1.61	2.21	2.13	0.80	1.39	0.98

Temp.	186	186	186	188	188	189	190	192
Ratio	1.77	1.96	2.74	1.43	2.50	2.98	1.87	3.10

(a) Determine the equation of the estimated regression line. (Round all numerical values to five decimal places.)
y =

(b) Calculate a point estimate for true average efficiency ratio when tank temperature is 186. (Round your answer to four decimal places.)


(c) Calculate the values of the residuals from the least squares line for the four observations for which temperature is 186. (Round your answers to four decimal places.)

(186, 0.98)
(186, 1.77)
(186, 1.96)
(186, 2.74)

(d) What proportion of the observed variation in efficiency ratio can be attributed to the simple linear regression relationship between the two variables? (Round your answer to four decimal places.)

Answer 1

Independent variable x: Temp

Dependent variable: y: Ratio

(a)

Following table shows the calculations:

	X	Y	X^2	Y^2	XY
	174	0.84	30276	0.7056	146.16
	176	1.33	30976	1.7689	234.08
	177	1.52	31329	2.3104	269.04
	178	1.11	31684	1.2321	197.58
	178	1.05	31684	1.1025	186.9
	179	1.14	32041	1.2996	204.06
	180	1.1	32400	1.21	198
	181	1.74	32761	3.0276	314.94
	184	1.41	33856	1.9881	259.44
	184	1.5	33856	2.25	276
	184	1.61	33856	2.5921	296.24
	184	2.21	33856	4.8841	406.64
	184	2.13	33856	4.5369	391.92
	185	0.8	34225	0.64	148
	185	1.39	34225	1.9321	257.15
	186	0.98	34596	0.9604	182.28
	186	1.77	34596	3.1329	329.22
	186	1.96	34596	3.8416	364.56
	186	2.74	34596	7.5076	509.64
	188	1.43	35344	2.0449	268.84
	188	2.5	35344	6.25	470
	189	2.98	35721	8.8804	563.22
	190	1.87	36100	3.4969	355.3
	192	3.1	36864	9.61	595.2
Total	4404	40.21	808638	77.2047	7424.41

(b)

(C)

Residual = observed -predicted

For (186, 0.98):

Residual = 0.98 - 1.90297 = -0.92297

For (186, 1.77):

Residual = 1.77 - 1.90297 = -0.13297

For (186, 1.96):

Residual = 1.96 - 1.90297 = 0.05703

For (186, 2.74):

Residual = 2.74 - 1.90297 = 0.83703

(D)

Answer: The 0.4251 proportion of the observed variation in efficiency ratio can be attributed to the simple linear regression relationship between the two variables.