Chapter 4.2, Problem 25E

Foot temperatures: Foot ulcers are a common problem for people with diabetes. Higher skin temperatures on the foot indicate an increased risk of ulcers. In a study carried out at the Colorado School of Mines, skin temperatures on both feet were measured, in degrees Fahrenheit, for 18 diabetic patients. The results are presented in the following table.

1. Compute due least-squares regression line for predicting right foot temperature from the left foot temperature.
2. Construct a scatter-plot (y) versus the left foot temperature Graph the least-squares regression line on the same axes.
3. If the left foot temperatures of two patients differ by 2 degrees: by how much would you predict their right foot temperatures to differ?
4. Predict the right foot temperature for a patient whose left foot temperature is 81 degrees.

To determine

To find: The least-square regression line for the given data set.

Answer to Problem 25E

The least square regression line of the given data set is,

  $\widehat{y}=25.0616-1.5881x$

Explanation of Solution

The foot temperature of both foots are measured of $18$ patients for a study to determine the risk of foot ulcers. Following table shows the measurements in Fahrenheit.

  $\begin{array}{cccc}\text{Left foot}& \text{Right foot}& \text{Left foot}& \text{Right foot}\\ 80& 80& 76& 81\\ 85& 85& 89& 86\\ 75& 80& 87& 82\\ 88& 86& 78& 78\\ 89& 87& 80& 81\\ 87& 82& 87& 82\\ 78& 78& 86& 85\\ 88& 89& 76& 80\\ 89& 90& 88& 89\end{array}$

Calculation:

The least-square regression is given by the formula,

  $\widehat{y}=b_0+b_{1}x$

Where $b_1=r\frac{s_y}{s_x}$ and $b_0=\overline{y}-b_1\overline{x}$

  $r$ is the correlation coefficient.

  $s_x$ is the standard deviation of $x$ .

  $s_y$ is the standard deviation of $y$ .

The correlation coefficient is given by the formula,

  $r=\frac{1}{n-1}{\displaystyle \sum \left(\frac{x-\overline{x}}{s_x}\right)\left(\frac{y-\overline{y}}{s_y}\right)}$

Supposing the variable $x$ is the temperature of left foot and $y$ is the temperature of right foot, the means and the standard deviations should be calculated to find the slope and the intercept of the line.

  

Using the data above, Minitab, the correlation coefficient can be obtained by the following table.

  $\begin{array}{ccccc}x& y& \frac{x-\overline{x}}{s_x}& \frac{y-\overline{y}}{s_y}& \left(\frac{x-\overline{x}}{s_x}\right)\left(\frac{y-\overline{y}}{s_y}\right)\\ 80& 80& -0.7003& -0.8868& 0.6210\\ 85& 85& 0.2547& 0.4216& 0.1073\\ 75& 80& -1.6553& -0.8868& 1.4679\\ 88& 86& 0.8277& 0.6833& 0.5655\\ 89& 87& 1.0187& 0.9449& 0.9626\\ 87& 82& 0.6367& -0.3634& -0.2314\\ 78& 78& -1.0823& -1.4101& 1.5262\\ 88& 89& 0.8277& 1.4683& 1.2152\\ 89& 90& 1.0187& 1.7299& 1.7622\\ 76& 81& -1.4643& -0.6251& 0.9154\\ 89& 86& 1.0187& 0.6833& 0.6960\\ 87& 82& 0.6367& -0.3634& -0.2314\\ 78& 78& -1.0823& -1.4101& 1.5262\\ 80& 81& -0.7003& -0.6251& 0.4378\\ 87& 82& 0.6367& -0.3634& -0.2314\\ 86& 85& 0.4457& 0.4216& 0.1879\\ 76& 80& -1.4643& -0.8868& 1.2985\\ 88& 89& 0.8277& 1.4683& 1.2152\end{array}$

  ${\displaystyle \sum \left(\frac{x-\overline{x}}{s_x}\right)\left(\frac{y-\overline{y}}{s_y}\right)}=13.8109$

Hence, the correlation coefficient is,

  $\begin{array}{c}r=\frac{1}{18-1}\times 13.8109\\ =\frac{13.8109}{17}\\ r=0.8124\end{array}$

Then, the coefficient $b_1$ should be,

  $\begin{array}{c}{b}_1=r\frac{s_y}{s_x}\\ =0.8124\times \frac{3.8216}{5.2356}\\ b_1=0.5930\end{array}$

Therefore,

  $\begin{array}{c}{b}_0=\overline{y}-b_1\overline{x}\\ =83.3889-\left(0.5930\right)\times 83.6667\\ =83.3889-33.7754\\ b_0=49.6135\end{array}$

Conclusion:

The least square regression line is found to be,

  $\widehat{y}=49.6153+0.5930x$

To determine

To graph:The scatter plot for the given data.

Explanation of Solution

Graph:

The scatter plot for the given data can be constructed by considering the temperature of left foot as $x$ variable and the temperature of right foot as $y$ variable.

  

Interpretation:

Out of all these $18$ ordered pairs, two clusters can be identified mainly. One is located in left-down of the plot and the other one is located right-up. In the middle area of the plot, no point can be observed.

To determine

To find:The temperature change of right foot for a change of $2$ Fahrenheit in the temperature of left foot.

Answer to Problem 25E

An increase of $1.1860$ Fahrenheit in the temperature of right foot can be expected for a $2$ Fahrenheit change of the temperature in left foot.

Explanation of Solution

Given:

The least-square regression line has been obtained as $\widehat{y}=49.6153+0.5930x$ . This formula can be used to predict the values of $y$ for a certain $x$ value.

Calculation:

Let $x_0$ be the left foot temperature of a particular patient. Then $x_0+2$ is the amount that is larger than $x_0$ by $2$ .

Then the corresponding prediction for the right foot temperature for $x_0$ can be obtained as,

  ${\widehat{y}}_{x_0}=49.6153+0.5930x_0$

Also, for $x_0+2$ ,

  ${\widehat{y}}_{x_0+2}=49.6153+0.5930\left(x_0+2\right)$

The second relationship can be simplified as follows.

  $\begin{array}{c}{\widehat{y}}_{x_0+2}=49.6153+0.5930\left(x_0+2\right)\\ =\underset{{\widehat{y}}_{x_0}}{\underbrace{49.6153+0.5930x_0}}+0.5930\times 2\\ {\widehat{y}}_{x_0+0.2}={\widehat{y}}_{x_0}+1.1860\end{array}$

The difference of these two predicted values gives the change of the right foot temperature for $2$ Fahrenheit change in the temperature of left foot.

  ${\widehat{y}}_{x_0+10}-{\widehat{y}}_{x_0}=1.1860$

Conclusion:

Therefore, an increase of $1.1860$ Fahrenheit in the temperature of right foot can be expected for a $2$ Fahrenheit change of the temperature in left foot.

To determine

To find: The predicted right foot temperature when the left foot is in $81$ degrees of Fahrenheit.

Answer to Problem 25E

The predicted right foot temperature is $97.6483$ degrees of Fahrenheit.

Explanation of Solution

Given:

The formula of the least square regression has been determined as $\widehat{y}=49.6153+0.5930x$

Calculation:

When the temperature of the left foot is $81$ Fahrenheit, the variable $x$ should be assigned as $x=81$ . By substituting this property into the formula, we can calculate the corresponding $y$ value. This is the predicted right foot temperature.

  $\begin{array}{c}\widehat{y}=49.6153+0.5930\times 81\\ =49.6153+48.033\\ \widehat{y}=97.6483\end{array}$

Conclusion:

Therefore, the predicted right foot temperature is $97.6483$ degrees.