Question
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.
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.
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.
Answers
GPT-4o mini
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:
\[
Y = 0.4311X + 48.1115
\]
\[
y = mx + b
\]
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:
\[
Y = 0.4311X + 48.1115
\]
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.
GPT-4o mini
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**.
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**.
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
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
GPT-4o mini
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**.
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**.
Predict the right foot temperature for a patient whose left foot temperature is 77 degrees. Round the answer to one decimal place.
GPT-4o mini
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**.
\[
y = 31.005 + 0.6253x
\]
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**.
One of your digits is wrong, please check your math
GPT-4o mini
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.
### 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.
that is still incorrect
GPT-4o mini
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.
### 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.