Chapter 8, Problem 1SE

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

$\text{Elongation}={\beta }_0+{\beta }_1\text{Oxygen}+{\beta }_2\text{Nitrogen}+{\beta }_3\text{Oxygen}\cdot \text{Nitrogen}$

1. a. Predict the elongation for a weld with an oxygen content of 0.15% and a nitrogen content of 0.01%.
2. 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?
3. 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.

To determine

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 $\text{Elongation}={\beta }_0+{\beta }_1\text{Oxygen+}{\beta }_2\text{Nitrogen}+{\beta }_3\text{Oxygen}\text{.Nitrogen}$ in order to determine the effect of oxygen content and nitrogen content in elongation percent of titanium welds.

Multiple linear regression model:

A multiple linear regression model is given as $y_i={\beta }_0+{\beta }_1{x}_{1i}+\mathrm{...}+{\beta }_k{x}_{ki}+{\epsilon }_i$ where $y_i$ is the response variable, and $x_{1i},x_{2i},\mathrm{...},x_{ki}$ are the k predictor variables. The quantities ${\beta }_0,{\beta }_1,\mathrm{...},{\beta }_k$ are the slopes corresponding to $x_{1i},x_{2i},\mathrm{...},x_{ki}$ respectively.${\widehat{\beta }}_0$ is the estimated intercept of the line, from the sample data.

The ‘Coefficient’ column of the regression analysis MINITAB output gives the slopes corresponding to the respective variables stored in the column ‘Predictor’.

Let $x_1,x_2$ be the Oxygen content % and Nitrogen content %.

From the accompanying MINITAB output, the intercept is $b_0=46.802$.

The estimates of the slopes are:

$b_1\left(\text{Oxygen}\right)=-130.11$

$b_2\left(\text{Nitrogen}\right)=-807.10$

$b_3\left(\text{Oxygen}\times \text{Nitrogen}\right)=3,580.5$

Thus, using the definition of a multiple regression model, the multiple regression equation is:

$\widehat{\text{Elongation}}=46.802-130.11\text{Oxygen}-807.10\text{Nitrogen}+3,580.5\text{Oxygen}\times \text{Nitrogen}$.

Here, $\text{Oxygen}\text{content}=0.15$ and $\text{Nitrogen}\text{content}=0.01$.

Predicted elongation percent of a weld:

$\begin{array}{c}\widehat{\text{Elongation}}=46.802-130.11\text{Oxygen}-807.10\text{Nitrogen}+3,580.5\text{Oxygen}\times \text{Nitrogen}\\ =46.802-\left(130.11\times 0.15\right)-\left(807.10\times 0.01\right)+\left(3,580.5\times 0.15\times 0.01\right)\\ =24.6\end{array}$

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%.

To determine

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 ${\beta }_i$ in a multiple regression equation is the amount of change in the response variable, $\widehat{y}$, due to unit increase in the corresponding predictor variable, $x_i$.

The multiple regression line is,

$\widehat{\text{Elongation}}=46.802-130.11\text{Oxygen}-807.10\text{Nitrogen}+3,580.5\text{Oxygen}\times \text{Nitrogen}$.

The coefficient or slope of Oxygen content in the regression model is $b_1\left(\text{Oxygen}\right)=-130.11$.

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,

$\begin{array}{c}0.05b_1+\left(0.05\times 0.006\times b_3\right)=0.05\times -130.11+\left(0.05\times 0.006\times 3,580.5\right)\\ =-6.5055+1.074\\ =-5.4315\end{array}$

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.

To determine

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 ${\beta }_i$ in a multiple regression equation is the amount of change in the response variable, $\widehat{y}$, due to unit increase in the corresponding predictor variable, $x_i$.

The multiple regression line is,

$\widehat{\text{Elongation}}=46.802-130.11\text{Oxygen}-807.10\text{Nitrogen}+3,580.5\text{Oxygen}\times \text{Nitrogen}$.

Here, the elongation is dependent on the nitrogen content, oxygen content and the interaction of nitrogen and oxygen content.

Hence, the coefficient of $\text{Oxygen}\times \text{Nitrogen}$ is also required to find the predicted change in elongation percent associated with 0.005% more oxygen content. Since, the value of the nitrogen content is necessary in the specified case.

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.