Chapter 13, Problem 66SE

a.

To determine

Identify and explain whichof the given modelscan be recommended.

Answer to Problem 66SE

The model with 2 predictors and the model with 3 predictors can be recommended for predicting the pH before addition of dyes.

Explanation of Solution

Given info:

The MINITAB output shows the best regression option for the data predicted for pH before the addition of dyes using carpet density, carpet weight, dye weight, dye weight as a percentage of carpet and pH after addition of dyes.

Justification:

Mallows ${\text{C}}_{\text{P}}$statistic:

It is used to assess the fit of regression model where the aim to find the best subset of predictors. A relatively small value of ${\text{C}}_{\text{P}}$ tells that the model is relatively precise.

By observing the mallows ${\text{C}}_{\text{P}}$ statistic it can be observed that model with two variables $\left\{x_3,x_5\right\}$ could be considered as a best model because the $R^2$,adjusted $R_a^2$ for this model is 68.7, 68.1 and its ${\text{C}}_{\text{P}}$ value is 1.2 which is the minimum value when compared to other models involving two variables.

By examining the models with three variables, $\left\{x_2,x_3,x_5\right\}$ could be considered as a best model because the $R^2$, adjusted $R_a^2$ for this model is 69, 68.2 and its ${\text{C}}_{\text{P}}$ value is 2.2 which is the minimum value when comparing to other models involving three variables.

Hence, the model with two predictorsnamely dye weight and pH after addition of dyes could be considered as a best model subset for predicting pH before the addition of dyes.

Also, a second option would be the model with three predictorsnamely carpet weight, dye weight and pH after addition of dyes could be considered as a best model subset for predicting pH before the addition of dyes.

b.

To determine

Test whether the model suggests a useful linear relationship between pH before the addition of dyes and at least one of the predictors.

Answer to Problem 66SE

There is sufficient evidence to conclude that the there is a use of linear relationship between pH before the addition of dyes and at least one of the predictors dye weight and pH after the addition of dyes.

Explanation of Solution

Given info:

The MINITAB output for predicting the pH before the addition of dyes using the dye weight $x_3$ and pH after the addition of dyes $x_5$ is given.

Calculation:

The test hypotheses are given below:

Null hypothesis:

$H_0:{\beta }_3={\beta }_5=0$

That is, there is no use of linear relationship between pH before the addition of dyes and the predictors dye weightand pH after the addition of dyes.

Alternative hypothesis:

$H_a:\text{At least one of the }\beta \text{'}\text{s}\ne 0$

That is, there is a use of linear relationship between pH before the addition of dyes and at least one of the predictors dye weightand pH after the addition of dyes.

Conclusion:

The P-value is 0.000 and the level of significance is 0.001.

The P-value is lesser than the level of significance.

That is $0.000\left(=P\text{-value}\right)<0.001\left(=\alpha \right)$.

Thus, the null hypothesis is rejected.

Hence, there is sufficient evidence to conclude that there is a use of linear relationship between pH before the addition of dyes and at least one of the predictors dye weight and pH after the addition of dyes.

c.

To determine

Explain whether either one of the predictors could be eliminated from the model given that the other predictor is retained.

Answer to Problem 66SE

No, either one of the predictors could not be eliminated from the model given that the other predictor is retained.

Explanation of Solution

Calculation:

For variable $x_3$:

Testing the hypothesis:

Null hypothesis:

$H_0:{\beta }_3=0$

That is, there is no use of linear relationship between pH before the addition of dyes and dye weightgiven that pH after addition of dyes was retained in the model.

Alternative hypothesis:

$H_a:{\beta }_3\ne 0$

That is, there is a use of linear relationship between pH before the addition of dyes and dye weightgiven that pH after addition of dyes was retained in the model.

From the MINITAB output it can be observed that the P-value corresponding to the t statistic of $x_3$ is 0.000.

Conclusion:

The P-value is 0.000 and the level of significance is 0.001.

The P-value is lesser than the level of significance.

That is $0.000\left(=P\text{-value}\right)<0.001\left(=\alpha \right)$.

Thus, the null hypothesis is rejected.

Hence, there is sufficient evidence to conclude that there is a use of linear relationship between pH before the addition of dyes and dye weight given that pH after addition of dyes was retained in the model.

For variable $x_5$:

Testing the hypothesis:

Null hypothesis:

$H_0:{\beta }_5=0$

That is, there is no use of linear relationship between pH before the addition of dyes and pH after addition of dyes given that dye weight was retained in the model.

Alternative hypothesis:

$H_a:{\beta }_3\ne 0$

That is, there is a use of linear relationship between pH before the addition of dyes and pH after addition of dyes given that dye weight was retained in the model.

From the MINITAB output it can be observed that the P-value corresponding to the t statistic of $x_5$ is 0.000.

Conclusion:

The P-value is 0.000 and the level of significance is 0.001.

The P-value is lesser than the level of significance.

That is $0.000\left(=P\text{-value}\right)<0.001\left(=\alpha \right)$.

Thus, the null hypothesis is rejected.

Hence, there is sufficient evidence to conclude that there is a use of linear relationship between pH before the addition of dyes and pH after addition of dyes given that dye weight was retained in the model.

Justification:

From the analysis it can be concluded that none of the variables can be eliminated from the model given that the other variable is already present in the model.

d.

To determine

Calculate and interpret the 95% confidence interval for the two predictors.

Answer to Problem 66SE

The 95% confidence interval for the estimated slope coefficient ${\widehat{\beta }}_3$ of dye weight is

(–0.0000684, –0.0000244).

The 95% confidence interval for the estimated slope coefficient ${\widehat{\beta }}_5$ of pH after the addition of dyes is (0.6417, 0.8325).

Explanation of Solution

Calculation:

The 95% confidence interval is calculated using the formula:

The confidence interval is calculated using the formula:

${\widehat{\beta }}_i\pm t_{\frac{\alpha }{2},n-\left(k+1\right)}{s}_{{\widehat{\beta }}_i}$

Where,

${\widehat{\beta }}_i$ is the estimated slope coefficient.

$\alpha$ is the level of significance.

n is the total number of observations.

k is the total number of predictors in the model.

$s_{{\widehat{\beta }}_i}$ is the standard error while calculating the estimated slope coefficient.

Critical value:

Software procedure:

Step-by-step procedure to find the critical value is given below:

• Click on Graph, select View Probability and click OK.
• Select t, enter 111 as Degrees of freedom, inShaded Area Tab select Probability under Define Shaded Area By and choose Both tails.
• Enter Probability value as 0.05.
• Click OK.

Output obtained from MINITAB is given below:

The 95% confidence interval for ${\widehat{\beta }}_3$ is given below:

$\begin{array}{c}{\widehat{\beta }}_3\pm t_{\frac{\alpha }{2},n-\left(k+1\right)}{s}_{{\widehat{\beta }}_3}=-0.0000464\pm t_{\frac{0.05}{2},114-\left(2+1\right)}\left(0.000011\right)\\ =-0.0000464\pm t_{0.025,111}\left(0.000011\right)\\ =-0.0000464\pm \left(1.982\right)\left(0.000011\right)\\ =-0.0000464\pm 0.000022\end{array}$

$=-0.0000684,-0.0000244$

Thus, the 95% confidence interval for the estimated slope coefficient ${\widehat{\beta }}_3$ is

(–0.0000684, –0.0000244).

The 95% confidence interval for ${\widehat{\beta }}_3$ is given below:

$\begin{array}{c}{\widehat{\beta }}_5\pm t_{\frac{\alpha }{2},n-\left(k+1\right)}{s}_{{\widehat{\beta }}_5}=0.073710\pm t_{\frac{0.05}{2},114-\left(2+1\right)}\left(0.04813\right)\\ =0.073710\pm t_{0.025,111}\left(0.04813\right)\\ =0.073710\pm \left(1.982\right)\left(0.04813\right)\\ =0.073710\pm 0.0954\end{array}$

$=0.6417,0.8325$

Thus, the 95% confidence interval for the estimated slope coefficient ${\widehat{\beta }}_5$ is

(0.6417,0.8325).

Interpretation:

For the variable $x_3$ dye weight:

For one unit increase in the dye weight, it is 95% confident that the estimated value of pH before addition of dyes would decrease between–0.00000684 and–0.0000244 given that pH after addition of dyes is fixed constant.

For the variable $x_5$pH after the addition of dyes:

For one unit increase in the pH after the addition of dyes it is 95% confident that the estimated value of pH before addition of dyes would increase between 0.6417 and 0.8325 given that dye weight is fixed constant.

e.

To determine

Calculate and interpret the 95% confidence interval for the average value of pH before the addition of dyes when the dye weight and pH after the addition of dyes takes 1,000 and 6, respectively.

Answer to Problem 66SE

The 95% confidence interval for the average value of pH before the addition of dyes when the dye weight and pH after the addition of dyes takes 1,000 and 6, respectively is (5.250, 5.383)

Explanation of Solution

Given info:

The estimated standard deviation for predicting the pH before the addition of dyes when the dye weight and pH after the addition of dyes takes 1,000 and 6 is 0.0336.

Calculation:

The average value of pH before the addition of dyes when the dye weight and pH after the addition of dyes takes 1,000 and 6 is calculated as follows:

$\begin{array}{c}\widehat{y}=0.9402-0.0000464x_3+0.73710x_5\\ =0.9402-0.0000464\left(1,000\right)+0.73710\left(6\right)\\ =0.9402-0.0464+4.4226\\ =5.316\end{array}$

Thus, the average value of pH before the addition of dyes when the dye weight and pH after the addition of dyes takes 1,000 and 6 is 5.316.

95% confidence interval for the true response:

The confidence interval is calculated using the formula:

${\widehat{Y}}_i\pm t_{\frac{\alpha }{2},n-\left(k+1\right)}{s}_{{\widehat{Y}}_i}$

Where,

${\widehat{Y}}_i$ is the estimated value of the dependent variable.

$\alpha$ is the level of significance.

n is the total number of observations.

k is the total number of predictors in the model.

$s_{{\widehat{Y}}_{\cdot i}}$ is the standard error while calculating the estimated value of the dependent variable.

Critical value:

Software procedure:

Step-by-step procedure to find the critical value is given below:

• Click on Graph, select View Probability and click OK.
• Select t, enter 111 as Degrees of freedom, in Shaded Area Tab select Probability under Define Shaded Area By and choose Both tails.
• Enter Probability value as 0.05.
• Click OK.

Output obtained from MINITAB is given below:

The 95% confidence interval is given below:

$\begin{array}{c}{\widehat{Y}}_i\pm t_{\frac{\alpha }{2},n-\left(k+1\right)}{s}_{{\widehat{Y}}_i}=5.316\pm t_{\frac{0.05}{2},114-\left(2+1\right)}\left(0.0336\right)\\ =5.316\pm t_{0.025,111}\left(0.0336\right)\\ =5.316\pm \left(1.982\right)\left(0.0336\right)\\ =5.316\pm 0.0666\end{array}$

$=5.250,5.383$

Thus, the 95% confidence interval for the average value of pH before the addition of dyes when the dye weight and pH after the addition of dyes takes 1,000 and 6 is (5.250,5.383).

Interpretation:

It is 95% confident that average value of pH before the addition of dyes when the dye weight and pH after the addition of dyes takes 1,000 and 6 would lie between 5.250 and 5.383.