Question
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.
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.
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.
Answers
GPT-4o mini
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**.
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**.
Predict the right foot temperature for a patient whose left foot temperature is 86 degrees. Round the answer to one decimal place.
GPT-4o mini
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**.
\[
y = 24.5 + 0.7083x
\]
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**.