The article “Advances in Oxygen Equivalence liquations for Predicting the Properties of Titanium Welds” (D. Harwig, W. Ittiwattana, and H. Castner, The Welding Journal, 2001:126s–136s) reports an experiment to predict various properties of titanium welds. Among other properties, the elongation (in %) was measured, along with the oxygen content and nitrogen content (both in percent). The following MINITAB output presents results of fitting the model
- a. Predict the elongation for a weld with an oxygen content of 0.15% and a nitrogen content of 0.01%.
- b. If two welds both have a nitrogen content of 0.006%, and their oxygen content differs by 0.05%, what would you predict their difference in elongation to be?
- c. Two welds have identical oxygen contents, and nitrogen contents that differ by 0.005%. Is this enough information to predict their difference in elongation? If so, predict the elongation. If not, explain what additional information is needed.
a.
Find the predicted elongation percent of a weld with 0.15% of oxygen content and 0.01% of nitrogen content.
Answer to Problem 1SE
The predicted elongation percent of a weld with 0.15% of oxygen content and 0.01% of nitrogen content is likely to be 24.6%.
Explanation of Solution
Calculation:
The data represents the MINITAB output of the regression model
Multiple linear regression model:
A multiple linear regression model is given as
The ‘Coefficient’ column of the regression analysis MINITAB output gives the slopes corresponding to the respective variables stored in the column ‘Predictor’.
Let
From the accompanying MINITAB output, the intercept is
The estimates of the slopes are:
Thus, using the definition of a multiple regression model, the multiple regression equation is:
Here,
Predicted elongation percent of a weld:
Thus, the predicted elongation percent of a weld with 0.15% of oxygen content and 0.01% of nitrogen content is likely to be 24.6%.
b.
Find the change between the elongation percent of the two welds when the nitrogen content is 0.006% for both the welds with one weld containing 0.05% more oxygen content.
Answer to Problem 1SE
The elongation percent of two welds differ by –5.43% when the nitrogen content is 0.006% for both the welds with one weld containing 0.05% more oxygen content.
Explanation of Solution
Justification:
Slope in a multiple regression equation:
The slope
The multiple regression line is,
The coefficient or slope of Oxygen content in the regression model is
From this it can be said that, the value of elongation percent decreases by 130.11 for a 1% increase in Oxygen content, provided the effects of Nitrogen content is accounted for.
Here, both the welds have same Nitrogen content 0.006% and one weld has 0.05% more oxygen content than the other.
The change between the elongation percent of two welds is,
Thus, the elongation percent of two welds differ by –5.43% when the nitrogen content is 0.006% for both the welds with one weld containing 0.05% more oxygen content.
c.
Check whether it is possible to estimate the change in the elongation percent of the two welds when the nitrogen content is same for both the welds with one weld containing 0.005% more oxygen content.
If possible, predict the change.
Answer to Problem 1SE
No, it is not possible to estimate the change in the elongation percent of the two welds when the nitrogen content is same for both the welds with one weld containing 0.005% more oxygen content.
Explanation of Solution
Justification:
Slope in a multiple regression equation:
The slope
The multiple regression line is,
Here, the elongation is dependent on the nitrogen content, oxygen content and the interaction of nitrogen and oxygen content.
Hence, the coefficient of
Therefore, it is not possible to determine the change in the elongation percent only with the value of oxygen content.
Thus, it is not possible to estimate the change in the elongation percent of the two welds when the nitrogen content is same for both the welds with one weld containing 0.005% more oxygen content.
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