Chapter 5.4, Problem 59E

a.

To determine

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

Answer to Problem 59E

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

Explanation of Solution

Calculation:

Reference: Exercise 5.58: The data on success (%), y and energy of shock, x is given.

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

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 ${x^{\prime }}^{\prime }=\ln \left(x\right)$:

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 LN(‘x’) under Expression.
• Click OK.

The transformed variable is stored in the column ln(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 ln(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 ${x^{\prime }}^{\prime }=\ln \left(x\right)$ 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 $\underset{\_}{{x^{\prime }}^{\prime }=\ln \left(x\right)}$ is recommended.

b.

To determine

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

Answer to Problem 59E

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 $\underset{\_}{\widehat{y}=62.42+43.89\ln \left(x\right)}$.

Explanation of Solution

Calculation:

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 ln(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: $\underset{\_}{\widehat{y}=62.42+43.89\ln \left(x\right)}$.

c.

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 59E

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:

$\begin{array}{c}\widehat{y}=62.42+43.89\ln \left(x\right)\\ =62.42+43.89\ln \left(1.75\right)\\ \approx 62.42+\left(43.89\times 0.5596\right)\\ \approx 62.42+24.56\\ =\mathrm{86.98.}\end{array}$

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:

$\begin{array}{c}\widehat{y}=62.42+43.89\ln \left(x\right)\\ =62.42+43.89\ln \left(0.8\right)\\ \approx 62.42+\left(43.89\times \left(-0.2231\right)\right)\\ \approx 62.42-9.79\\ =\mathrm{52.63.}\end{array}$

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