oot ulcers are a common problem for people with diabetes. Higher skin temperatures on the foot indicate an increased risk of ulcers. In a study performed at the Colorado School of Mines, skin temperatures on both feet were measured, in degrees Fahrenheit, for 8 diabetic patients.

Question

Left foot
78
89
80
87
88
85
89
76
Right Foot
78
87
80
82
89
85
90
81

The regression line is y=24.5+0.7083x

If the left foot temperatures of two patients differ by 3 degrees, by how much would you predict their right foot temperatures to differ? Round the answer to three decimal places.

Answer 1

To find out how much the right foot temperatures would differ based on a 3-degree difference in left foot temperatures, we can use the slope of the regression line.

The regression line given is: \[ y = 24.5 + 0.7083x \] where \( y \) is the predicted right foot temperature and \( x \) is the left foot temperature.

The slope \( 0.7083 \) indicates how much the right foot temperature changes for each 1-degree change in left foot temperature.

Given that the left foot temperatures of two patients differ by 3 degrees, we can calculate the predicted difference in the right foot temperatures (denoted as \( \Delta y \)) as follows:

\[ \Delta y = 0.7083 \times \Delta x \] where \( \Delta x = 3 \) degrees.

Now substituting the value:

\[ \Delta y = 0.7083 \times 3 = 2.1249 \]

Now we’ll round the result to three decimal places:

\[ \Delta y \approx 2.125 \]

Thus, if the left foot temperatures of two patients differ by 3 degrees, we would predict their right foot temperatures to differ by approximately 2.125 degrees.

Answer 2

Predict the right foot temperature for a patient whose left foot temperature is 86 degrees. Round the answer to one decimal place.

Answer 3

To predict the right foot temperature for a patient whose left foot temperature is 86 degrees, we will use the regression line equation:

\[ y = 24.5 + 0.7083x \]

where \( x \) is the left foot temperature. In this case, \( x = 86 \).

Now, substituting the value of \( x \) into the equation:

\[ y = 24.5 + 0.7083 \times 86 \]

First, calculate the multiplication:

\[ 0.7083 \times 86 = 60.918 \]

Now, add this to 24.5:

\[ y = 24.5 + 60.918 = 85.418 \]

Now we round the answer to one decimal place:

\[ y \approx 85.4 \]

Thus, the predicted right foot temperature for a patient whose left foot temperature is 86 degrees is approximately 85.4 degrees.