Chapter 5, Problem 18CRE

a.

To determine

Draw a scatterplot for the given data and identify the pattern as linear or nonlinear.

Answer to Problem 18CRE

The scatterplot for the data is obtained as follows:

The relationship between the variables is nonlinear.

Explanation of Solution

Calculation:

The data on success (%), y and energy of shock, x is given.

Scatterplot:

Software procedure:

Step-by-step procedure to draw the scatterplot using MINITAB software is given below:

• Choose Graph > Scatterplot.
• Choose Simple, and then click OK.
• Enter the column of y under Y variables.
• Enter the column of x under X variables.
• Click OK.

Thus, the scatterplot is obtained.

A careful observation of the scatterplot reveals that for lower values of x, the points are close to being linear. However, the curvature in the distribution of the points gradually increases with increasing values of x.

Thus, the relationship between the variables is nonlinear.

b.

To determine

Fit a least-squares regression line to the data.

Construct a residual plot for the model.

Explain whether the residual plot supports the conclusion in Part a.

Answer to Problem 18CRE

The least-squares regression line for the data is $\underset{\_}{\widehat{y}=22.48+34.36x}$.

Explanation of Solution

Calculation:

The least-squares regression line can be obtained using software.

Regression:

Software procedure:

Step by step procedure to get regression equation using MINITAB software is given as,

• Choose Stat > Regression > Regression > Fit Regression Model.
• Under Responses, enter the column of y.
• Under Continuous predictors, enter the columns of x.
• Choose Results and select Analysis of Variance, Model Summary, Coefficients, Regression Equation.
• Choose Graphs, under Residual versus the variables, enter x.
• Click OK on all dialogue boxes.

The outputs using MINITAB software is given as follows:

Regression analysis:

Residual plot:

From the output, the least-squares regression line is: $\underset{\_}{\widehat{y}=22.48+34.36x}$.

The ideal residual plot for a linear regression model must not show any pattern and must be randomly distributed. However, this residual plot clearly shows a curved pattern, with an approximate inverted U-shape. This suggests that the data is not linearly distributed.

Thus, the residual plot supports the conclusion in Part a.

c.

To determine

Justify whether the transformation $x^{\prime }=\sqrt{x}$ or ${x^{\prime }}^{\prime }=\log \left(x\right)$ is recommended, by keeping y unchanged.

Answer to Problem 18CRE

The transformation $\underset{\_}{{x^{\prime }}^{\prime }=\log \left(x\right)}$ is recommended.

Explanation of Solution

Calculation:

The suitable transformation can be identified by constructing scatterplot between y and $x^{\prime }=\sqrt{x}$, and between y and ${x^{\prime }}^{\prime }=\log \left(x\right)$.

Consider the transformed variable, $x^{\prime }$ as $\text{sqrt}\left(x\right)=\sqrt{x}$. The transformation can be obtained using software.

Data transformation $\text{sqrt}\left(x\right)=\sqrt{x}$:

Software procedure:

Step-by-step procedure to transform the data using MINITAB software is given below:

• Choose Calc > Calculator.
• Enter the column of sqrt(x) under Store result in variable.
• Enter the formula SQRT(‘x’) under Expression.
• Click OK.

The transformed variable is stored in the column sqrt(x).

Scatterplot:

Software procedure:

Step-by-step procedure to draw the scatterplot using MINITAB software is given below:

• Choose Graph > Scatterplot.
• Choose Simple, and then click OK.
• Enter the column of y under Y variables.
• Enter the column of sqrt(x) under X variables.
• Click OK.

The output obtained using MINITAB is as follows:

Consider the transformed variable, $x^{\prime }$ as $\text{sqrt}\left(x\right)=\sqrt{x}$.

Data transformation ${x^{\prime }}^{\prime }=\log \left(x\right)$:

Software procedure:

Step-by-step procedure to transform the data using MINITAB software is given below:

• Choose Calc > Calculator.
• Enter the column of sqrt(x) under Store result in variable.
• Enter the formula LOGTEN(‘x’) under Expression.
• Click OK.

The transformed variable is stored in the column log(x).

Scatterplot:

Software procedure:

Step-by-step procedure to draw the scatterplot using MINITAB software is given below:

• Choose Graph > Scatterplot.
• Choose Simple, and then click OK.
• Enter the column of y under Y variables.
• Enter the column of log(x) under X variables.
• Click OK.

The output obtained using MINITAB is as follows:

A careful observation of the scatterplot between y and $x^{\prime }=\sqrt{x}$ shows that the points form an extended S-shaped pattern. This does not suggest that a linear model would suitably describe the relationship between y and $x^{\prime }=\sqrt{x}$.

On the other hand, the scatterplot between y and ${x^{\prime }}^{\prime }=\log \left(x\right)$ shows a straightened plot. Although there is a slight curvature observed in this graph too, it is negligible. In comparison to the previous scatterplot, this appears to be much straighter.

Thus, the transformation $\underset{\_}{{x^{\prime }}^{\prime }=\log \left(x\right)}$ is recommended.

d.

To determine

Find the least-squares regression line between y and the transformation recommended in the previous part.

Answer to Problem 18CRE

The least-squares regression equation between y and the transformation recommended in the previous part, that is, ${x^{\prime }}^{\prime }=\log \left(x\right)$, is $\underset{\_}{\widehat{y}=62.42+101.06\ln \left(x\right)}$.

Explanation of Solution

Calculation:

The least-squares regression line can be obtained using software.

Regression:

Software procedure:

Step by step procedure to get regression equation using MINITAB software is given as,

• Choose Stat > Regression > Regression > Fit Regression Model.
• Under Responses, enter the column of y.
• Under Continuous predictors, enter the columns of log(x).
• Choose Results and select Analysis of Variance, Model Summary, Coefficients, Regression Equation.
• Click OK on all dialogue boxes.

The outputs using MINITAB software is given as follows:

From the output, the least-squares regression equation between y and the transformation recommended in the previous part, that is, ${x^{\prime }}^{\prime }=\ln \left(x\right)$, is: $\underset{\_}{\widehat{y}=62.42+101.06\ln \left(x\right)}$.

e.

To determine

Predict the success for an energy shock 1.75 times the threshold.

Predict the success for an energy shock 0.8 times the threshold.

Answer to Problem 18CRE

The success for an energy shock 1.75 times the threshold is 86.98%.

The success for an energy shock 0.8 times the threshold is 52.63%.

Explanation of Solution

Calculation:

The energy of shock is given as a multiple of the threshold of defibrillation.

For an energy shock 1.75 times the threshold, $x=1.75$. Substitute this value in the obtained regression equation:

$\begin{array}{c}\widehat{y}=62.42+101.06\log \left(x\right)\\ =62.42+101.06\log \left(1.75\right)\\ \approx 62.42+\left(101.06\times 0.2430\right)\\ \approx 62.42+24.56\\ =\mathrm{86.98.}\end{array}$

Thus, the success for an energy shock 1.75 times the threshold is 86.98%.

For an energy shock 0.8 times the threshold, $x=0.8$. Substitute this value in the obtained regression equation:

$\begin{array}{c}\widehat{y}=62.42+101.06\log \left(x\right)\\ =62.42+101.06\log \left(0.8\right)\\ \approx 62.42+\left(101.06\times \left(-0.0969\right)\right)\\ \approx 62.42-9.79\\ =\mathrm{52.63.}\end{array}$

Thus, the success for an energy shock 0.8 times the threshold is 52.63%.