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 8 diabetic patients. The results are presented in the following table.

Question

Left- 80, 75, 85, 88, 89, 78, 87, 88
Right- 80, 80, 85, 86, 87, 78, 82, 89
Compute the least-squares regression line for predicting the right foot temperature from the left foot temperature. Round the slope and y-intercept values to four decimal places.

Answer 1

To compute the least-squares regression line for predicting the right foot temperature from the left foot temperature, we need to use the formula for the regression line given by:

\[ y = mx + b \]

where:

• \(y\) is the predicted temperature of the right foot,
• \(x\) is the temperature of the left foot,
• \(m\) is the slope of the line,
• \(b\) is the y-intercept.

Step 1: Gather the data

Given:

• Left foot temperatures (\(X\)): 80, 75, 85, 88, 89, 78, 87, 88
• Right foot temperatures (\(Y\)): 80, 80, 85, 86, 87, 78, 82, 89

Step 2: Calculate means of \(X\) and \(Y\)

1. Calculate \(\bar{X}\) and \(\bar{Y}\):

\[ \bar{X} = \frac{(80 + 75 + 85 + 88 + 89 + 78 + 87 + 88)}{8} = \frac{ 80 + 75 + 85 + 88 + 89 + 78 + 87 + 88}{8} = \frac{ 80 + 75 + 85 + 88 + 89 + 78 + 87 + 88}{8} = \frac{ 680}{8} = 85 \]

\[ \bar{Y} = \frac{(80 + 80 + 85 + 86 + 87 + 78 + 82 + 89)}{8} = \frac{ 80 + 80 + 85 + 86 + 87 + 78 + 82 + 89}{8} = \frac{ 678}{8} = 84.75 \]

Step 3: Calculate the slope \(m\)

2. Use the formula:

\[ m = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sum{(X_i - \bar{X})^2}} \]

3. Calculate \(\sum{(X_i - \bar{X})(Y_i - \bar{Y})}\) and \(\sum{(X_i - \bar{X})^2}\):

• For each \(i\), calculate \(X_i - \bar{X}\), \(Y_i - \bar{Y}\), \((X_i - \bar{X})(Y_i - \bar{Y})\), and \((X_i - \bar{X})^2\).

| \(X_i\) | \(Y_i\) | \(X_i - \bar{X}\) | \(Y_i - \bar{Y}\) | \((X_i - \bar{X})(Y_i - \bar{Y})\) | \((X_i - \bar{X})^2\) | |---------|---------|--------------------|--------------------|------------------------------------|-----------------------| | 80 | 80 | -5 | -4.75 | 23.75 | 25 | | 75 | 80 | -10 | -4.75 | 47.5 | 100 | | 85 | 85 | 0 | 0.25 | 0 | 0 | | 88 | 86 | 3 | 1.25 | 3.75 | 9 | | 89 | 87 | 4 | 2.25 | 9 | 16 | | 78 | 78 | -7 | -6.75 | 47.25 | 49 | | 87 | 82 | 2 | -2.75 | -5.5 | 4 | | 88 | 89 | 3 | 4.25 | 12.75 | 9 |

• \(\sum{(X_i - \bar{X})(Y_i - \bar{Y})} = 23.75 + 47.5 + 0 + 3.75 + 9 - 5.5 + 12.75 = 91.5\)

• \(\sum{(X_i - \bar{X})^2} = 25 + 100 + 0 + 9 + 16 + 49 + 4 + 9 = 212\)

4. Calculate \(m\):

\[ m = \frac{91.5}{212} \approx 0.4311 \]

Step 4: Calculate the y-intercept \(b\)

5. Use the formula:

\[ b = \bar{Y} - m\bar{X} \]

\[ b = 84.75 - (0.4311 \times 85) \approx 84.75 - 36.6385 \approx 48.1115 \]

Step 5: Write the regression line

The least-squares regression line to predict the right foot temperature \(Y\) from the left foot temperature \(X\) is:

\[ Y = 0.4311X + 48.1115 \]

Final Result

Thus, rounding to four decimal places, the least-squares regression line is:

\[ Y = 0.4311X + 48.1115 \]

Answer 2

If the left foot temperatures of two patients differ by 4 degrees, by how much would you predict their right foot temperatures to differ? Round the answer to three decimal places.

Answer 3

To predict the difference in right foot temperatures based on a difference of 4 degrees in left foot temperatures, we can use the slope \(m\) of the least-squares regression line that we calculated earlier.

The slope \(m\) is the change in the predicted value of \(Y\) (right foot temperature) for a one-unit change in \(X\) (left foot temperature).

From the previous calculations, we found:

\[ m = 0.4311 \]

Now, if the left foot temperatures of two patients differ by 4 degrees, the predicted difference in right foot temperatures (\( \Delta Y \)) would be given by:

\[ \Delta Y = m \times \Delta X \]

Where \(\Delta X = 4\) degrees.

Substituting the values:

\[ \Delta Y = 0.4311 \times 4 \] \[ \Delta Y = 1.7244 \]

Now, rounding to three decimal places:

\[ \Delta Y \approx 1.724 \]

Thus, if the left foot temperatures of two patients differ by 4 degrees, we would predict their right foot temperatures to differ by approximately 1.724 degrees.

Answer 4

y= 31.005 + 0.6253x

If the left foot temperatures of two patients differ by 4 degrees, by how much would you predict their right foot temperatures to differ? Round the answer to three decimal places

Answer 5

To predict the difference in right foot temperatures based on a difference of 4 degrees in left foot temperatures, we can use the slope of the regression line provided.

From the equation:

\[ y = 31.005 + 0.6253x \]

The slope \( m \) is \( 0.6253 \). This means that for every 1-degree increase in left foot temperature (\( x \)), the right foot temperature (\( y \)) increases by \( 0.6253 \) degrees.

If the left foot temperatures of two patients differ by \( 4 \) degrees (\( \Delta x = 4 \)), the predicted difference in their right foot temperatures (\( \Delta y \)) can be calculated as follows:

\[ \Delta y = m \times \Delta x \]

Substituting the values:

\[ \Delta y = 0.6253 \times 4 \] \[ \Delta y = 2.5012 \]

Now rounding to three decimal places:

\[ \Delta y \approx 2.501 \]

Thus, if the left foot temperatures of two patients differ by 4 degrees, we would predict their right foot temperatures to differ by approximately 2.501 degrees.

Answer 6

Predict the right foot temperature for a patient whose left foot temperature is 77 degrees. Round the answer to one decimal place.

Answer 7

To predict the right foot temperature for a patient with a left foot temperature of 77 degrees using the provided regression equation:

\[ y = 31.005 + 0.6253x \]

we substitute \(x = 77\) into the equation:

\[ y = 31.005 + 0.6253 \times 77 \]

Now, performing the multiplication:

\[ 0.6253 \times 77 = 48.6761 \]

Next, we add this to the intercept:

\[ y = 31.005 + 48.6761 = 79.6811 \]

Now, rounding the answer to one decimal place:

\[ y \approx 79.7 \]

Thus, the predicted right foot temperature for a patient whose left foot temperature is 77 degrees is approximately 79.7 degrees.

Answer 8

One of your digits is wrong, please check your math

Answer 9

Let's go through the calculation step-by-step again to ensure the accuracy of the predicted right foot temperature using the regression equation:

Given Regression Equation

\[ y = 31.005 + 0.6253x \]

Substitute \(x = 77\)

\[ y = 31.005 + 0.6253 \times 77 \]

Calculate \(0.6253 \times 77\)

1. First, perform the multiplication: \[ 0.6253 \times 77 \]

   Calculating: \[ 0.6253 \times 77 = 48.5941 \]

Add this value to the intercept

Now, adding the result to the intercept: \[ y = 31.005 + 48.5941 = 79.5991 \]

Round the answer to one decimal place

\[ y \approx 79.6 \]

Thus, the corrected predicted right foot temperature for a patient whose left foot temperature is 77 degrees is approximately 79.6 degrees. Thank you for your patience, and I apologize for the error in my previous calculation.

Answer 10

that is still incorrect

Answer 11

Apologies for the oversight in my previous calculations. Let’s go through the calculations one more time, step by step, to ensure accuracy.

Given Regression Equation

\[ y = 31.005 + 0.6253x \]

Substitute \(x = 77\)

We want to find \(y\) when \(x = 77\): \[ y = 31.005 + 0.6253 \times 77 \]

Calculate \(0.6253 \times 77\)

Calculating that multiplication first: \[ 0.6253 \times 77 \] Breaking it down: \[ 0.6253 \times 77 = 48.6281 \]

Add this value to the intercept

Now we add this result to the intercept: \[ y = 31.005 + 48.6281 \] Calculating this sum: \[ y = 79.6331 \]

Round the answer to one decimal place

Rounding \(79.6331\) to one decimal place: \[ y \approx 79.6 \]

Thus, the correct predicted right foot temperature for a patient whose left foot temperature is 77 degrees is indeed approximately 79.6 degrees.

If there is a specific issue or point of confusion you notice, please let me know so we can address it together! Thank you for your patience.