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 performed at the Colorado School of Mines, skin temperatures on both feet were measured, in degrees Fahrenheit, for 10 diabetic patients. The results are presented in the following table.

Question

left
80
87
85
88
89
78
76
89
right
80
82
85
89
90
78
81
76

the regression line is y=26.5625+0.6823x

If the left foot temperatures of two patients differ by 5 degrees, by how much would you predict their right foot temperatures to differ? Round the answer to three decimal places.

Answer 1

To predict the difference in right foot temperatures based on the difference in left foot temperatures, we can use the slope from the given regression equation.

The regression line we're given is:

\[ y = 26.5625 + 0.6823x \]

In this equation, \( x \) represents the left foot temperature, and \( y \) represents the right foot temperature. The slope of the regression line, \( 0.6823 \), tells us how much the right foot temperature (y) is expected to change for a one-degree change in the left foot temperature (x).

If the left foot temperatures of two patients differ by 5 degrees, the corresponding difference in right foot temperatures can be calculated as follows:

\[ \text{Difference in right foot temperatures} = \text{Slope} \times \text{Difference in left foot temperatures} \]

Substituting the values into the equation:

\[ \text{Difference in right foot temperatures} = 0.6823 \times 5 \]

Calculating this:

\[ 0.6823 \times 5 = 3.4115 \]

Rounding to three decimal places, we have:

\[ \text{Predicted difference in right foot temperatures} \approx 3.412 \]

Thus, if the left foot temperatures of two patients differ by 5 degrees, we would predict their right foot temperatures to differ by approximately 3.412 degrees Fahrenheit.

Answer 2

Predict the right foot temperature for a patient whose left foot temperature is 82 degrees. Round the answer to one decimal place.

Answer 3

To predict the right foot temperature using the given regression line equation:

\[ y = 26.5625 + 0.6823x \]

where \( x \) is the left foot temperature, we can substitute \( x = 82 \) degrees into the equation:

\[ y = 26.5625 + 0.6823 \times 82 \]

Now, calculate \( 0.6823 \times 82 \):

\[ 0.6823 \times 82 = 55.5686 \]

Now, substitute this value back into the equation:

\[ y = 26.5625 + 55.5686 \]

Adding these values together:

\[ y = 82.1311 \]

Now, round the result to one decimal place:

\[ y \approx 82.1 \]

Thus, the predicted right foot temperature for a patient whose left foot temperature is 82 degrees is 82.1 degrees Fahrenheit.