The efficiency for a steel specimen immersed in a phosphating tank is the weight
ID: 412938 • Letter: T
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/ft. An article gave the accompanying data on tank temperature (x) and efficiency ratio (y) Temp. 172 174 175 176 176 177 178 179 Ratio 0.80 1.27 1.36 0.97 1.11 1.08 1.04 1.86 Temp. 182 182 182 182 182 183 183 184 Ratio 1.43 1.68 1.71 2.23 2.13 0.78 1.37 0.88 Temp. 184 184 184 186 186 187 188 190 Ratio 1.75 2.00 2.68 1.41 2.46 2.98 1.95 3.06 (a) Determine the equation of the estimated regression line. (Round all numerical values to five decimal places.) y0.09529 + 1.50748x X (b) Calculate a point estimate for true average efficiency ratio when tank temperature is 184. (Round your answer to four decimal places.) 19.0404 (c) Calculate the values of the residuals from the least squares line for the four observations for which temperature is 184. (Round your answers to four decimal places.) (184, 0.88) (184, 1.75) (184, 2.00) (184, 2.68) Why do they not all have the same sign? 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 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 cases of the first two pairs of observations, the observed efficiency ratios were smaller than the predicted value. In the case 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 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. (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.)Explanation / Answer
The given data is as follows:
Temperature (x)
Ratio (y)
172
0.8
174
1.27
175
1.36
176
0.97
176
1.11
177
1.08
178
1.04
179
1.86
182
1.43
182
1.68
182
1.71
182
2.23
182
2.13
183
0.78
183
1.37
184
0.88
184
1.75
184
2
184
2.68
186
1.41
186
2.46
187
2.98
188
1.95
190
3.06
The ratio value depends on temperature of solution. Thus, Temperature is independent variable (x) and ratio is dependent variable (y).
Using Regression analysis tool in Excel, the output obtained is as follows:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.668444
R Square
0.446817
Adjusted R Square
0.421672
Standard Error
0.50747
Observations
24
ANOVA
df
SS
MS
F
Significance F
Regression
1
4.576192
4.576192
17.76983
0.000357
Residual
22
5.665571
0.257526
Total
23
10.24176
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
Intercept
-15.6285
4.104027
-3.80808
0.000962
-24.1397
-7.11724
-24.1397
-7.11724
x
0.095288
0.022605
4.215427
0.000357
0.048409
0.142167
0.048409
0.142167
Regression Line equation is given as follows:
Y = a + bt
a = intercept = -15.6285
b = slope = 0.0953
Thus, the regression equation is y = -15.6285 + 0.0953x
a. y = -15.6285 + 0.0953x
b.
When, x = 184, y = -15.6285 + 0.0953(184) = 1.9044
Ratio when temperature is 184 = 1.9045
c.
Residual = Actual value y – calculated estimate of y
Observations
Actual Value of y
Estimate of y
Residual = Actual - Estimate
(184, 0.88)
0.88
1.9045
0.88-1.9045
= -1.02
(184, 1.75)
1.75
1.9045
-0.15
(814, 2.00)
2.00
1.9045
0.1
(184, 2.68)
2.68
1.9045
0.78
For the first two observation, the actual value of ratios are smaller than the estimate value and for last two observation the actual ratios are greater than estimated value
ANS: c
d.
Coefficient of Correlation (R), gives an idea of percentage of data points those fall within the results of the determined regression line. from regression output report R = 0.6844
Thus, for given data 68.44% of data points fall within the regression line
Temperature (x)
Ratio (y)
172
0.8
174
1.27
175
1.36
176
0.97
176
1.11
177
1.08
178
1.04
179
1.86
182
1.43
182
1.68
182
1.71
182
2.23
182
2.13
183
0.78
183
1.37
184
0.88
184
1.75
184
2
184
2.68
186
1.41
186
2.46
187
2.98
188
1.95
190
3.06
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