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.	171	173	174	175	175	176	177	178
Ratio	0.80	1.35	1.36	1.13	1.09	1.04	1.14	1.70

Temp.	181	181	181	181	181	182	182	183
Ratio	1.53	1.54	1.51	2.03	2.17	0.94	1.37	0.98

Temp.	183	183	183	185	185	186	187	189
Ratio	1.83	1.96	2.70	1.41	2.42	3.08	1.77	3.18

(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 183. (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 183. (Round your answers to four decimal places.)

(183, 0.98)
(183, 1.83)
(183, 1.96)
(183, 2.70)

Why do they not all have the same sign?

These residuals do not all have the same sign because in the cases of the first two pairs of observations, the observed efficiency ratios were smaller than the predicted value. In the cases of the last two pairs of observations, the observed efficiency ratios were larger than the predicted value.These residuals do not all have the same sign because in the cases of the first two pairs of observations, the observed efficiency ratios were larger than the predicted value. In the cases of the last two pairs of observations, the observed efficiency ratios were smaller than the predicted value.    These residuals do not all have the same sign because in the case of the third pair of observations, the observed efficiency ratio was equal to the predicted value. In the cases of the other pairs of observations, the observed efficiency ratios were smaller than the predicted value.These residuals do not all have the same sign because in the case of the second pair of observations, the observed efficiency ratio was equal to the predicted value. In the cases of the other pairs of observations, the observed efficiency ratios were larger than the predicted value.

(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

• Solution:

  Solution: We can use the excel regression data analysis tool to find the answer to the given questions. The excel output is given below:

• SUMMARY OUTPUT
  Regression Statistics
  Multiple R	0.673715
  R Square	0.453892
  Adjusted R Square	0.429068
  Standard Error	0.49527
  Observations	24
  ANOVA
  	df	SS	MS	F	Significance F
  Regression	1	4.485173	4.485173	18.28504	0.000307
  Residual	22	5.396423	0.245292
  Total	23	9.881596
  	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
  Intercept	-15.3596	3.983305	-3.856	0.000856	-23.6205	-7.09874	-23.6205	-7.09874
  Temp.	0.094335	0.022061	4.276101	0.000307	0.048583	0.140087	0.048583	0.140087

a) estimated regression line.

y = 15.35961 +0.09434*Temp

b) point estimate for true average efficiency ratio when tank temperature is 183

y = -15.35961+0.09434*183

=1.90461

c)

x	y	Predicted Ratio	Residuals
183	0.98	1.9046	-0.9246
183	1.83	1.9046	-0.0746
183	1.96	1.9046	0.0554
183	2.7	1.9046	0.7954

(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: 0.4539 = 45.89%

R Square

0.453891532