Asked by unf0rgettxble
The table below shows how a person's age is related to their medium income
Age: 26, 27, 28, 29, 30
Median income $1,000: 16.8, 19.1, 23.3, 25.8, 33.9
A. Find the equation for the line of best fit
B. Estimate the medium income for someone 32 years old
Age: 26, 27, 28, 29, 30
Median income $1,000: 16.8, 19.1, 23.3, 25.8, 33.9
A. Find the equation for the line of best fit
B. Estimate the medium income for someone 32 years old
Answers
Answered by
unf0rgettxble
Wouldn't a be 2.5 x
Answered by
unf0rgettxble
Use the point slope formula for a please no por favor
Answered by
GPT-4o mini
To find the equation for the line of best fit and to estimate the median income for someone who is 32 years old, we can follow these steps.
### A. Find the equation for the line of best fit
1. **Data Preparation:**
Let's denote Age as \( x \) and Median Income as \( y \). From the given data, we have the following points:
\[
(26, 16.8), (27, 19.1), (28, 23.3), (29, 25.8), (30, 33.9)
\]
2. **Calculating the necessary sums:**
To find the best fit line, we will use the least squares method. The equation of the line will be in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Number of points, \( n = 5 \)
- Sum of \( x \):
\[
\sum x = 26 + 27 + 28 + 29 + 30 = 140
\]
- Sum of \( y \):
\[
\sum y = 16.8 + 19.1 + 23.3 + 25.8 + 33.9 = 118.9
\]
- Sum of \( xy \):
\[
\sum xy = (26 \cdot 16.8) + (27 \cdot 19.1) + (28 \cdot 23.3) + (29 \cdot 25.8) + (30 \cdot 33.9) = 436.8 + 515.7 + 652.4 + 748.2 + 1017 = 3370.1
\]
- Sum of \( x^2 \):
\[
\sum x^2 = 26^2 + 27^2 + 28^2 + 29^2 + 30^2 = 676 + 729 + 784 + 841 + 900 = 3930
\]
3. **Calculating the slope \( m \):**
The formulas for the slope \( m \) and intercept \( b \) are:
\[
m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]
Plugging in the values:
\[
m = \frac{5(3370.1) - (140)(118.9)}{5(3930) - (140)^2}
\]
\[
= \frac{16850.5 - 16646}{19650 - 19600} = \frac{204.5}{50} = 4.09
\]
4. **Calculating the y-intercept \( b \):**
\[
b = \frac{\sum y - m \sum x}{n} = \frac{118.9 - 4.09(140)}{5}
\]
\[
= \frac{118.9 - 572.6}{5} = \frac{-453.7}{5} = -90.74
\]
5. **Final equation:**
Thus, the equation of the line of best fit is:
\[
y = 4.09x - 90.74
\]
### B. Estimate the median income for someone 32 years old
To estimate the median income for someone who is 32 years old, we can plug \( x = 32 \) into the equation of the line we found.
\[
y = 4.09(32) - 90.74
\]
\[
= 130.88 - 90.74
\]
\[
= 40.14
\]
### Conclusion
- **The equation for the line of best fit is:**
\[
y = 4.09x - 90.74
\]
- **The estimated median income for someone who is 32 years old is approximately:**
\[
\$40,140
\] (Note: Ensure consistency with units as the median income might be represented in a certain range based on context or scaling).
### A. Find the equation for the line of best fit
1. **Data Preparation:**
Let's denote Age as \( x \) and Median Income as \( y \). From the given data, we have the following points:
\[
(26, 16.8), (27, 19.1), (28, 23.3), (29, 25.8), (30, 33.9)
\]
2. **Calculating the necessary sums:**
To find the best fit line, we will use the least squares method. The equation of the line will be in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Number of points, \( n = 5 \)
- Sum of \( x \):
\[
\sum x = 26 + 27 + 28 + 29 + 30 = 140
\]
- Sum of \( y \):
\[
\sum y = 16.8 + 19.1 + 23.3 + 25.8 + 33.9 = 118.9
\]
- Sum of \( xy \):
\[
\sum xy = (26 \cdot 16.8) + (27 \cdot 19.1) + (28 \cdot 23.3) + (29 \cdot 25.8) + (30 \cdot 33.9) = 436.8 + 515.7 + 652.4 + 748.2 + 1017 = 3370.1
\]
- Sum of \( x^2 \):
\[
\sum x^2 = 26^2 + 27^2 + 28^2 + 29^2 + 30^2 = 676 + 729 + 784 + 841 + 900 = 3930
\]
3. **Calculating the slope \( m \):**
The formulas for the slope \( m \) and intercept \( b \) are:
\[
m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]
Plugging in the values:
\[
m = \frac{5(3370.1) - (140)(118.9)}{5(3930) - (140)^2}
\]
\[
= \frac{16850.5 - 16646}{19650 - 19600} = \frac{204.5}{50} = 4.09
\]
4. **Calculating the y-intercept \( b \):**
\[
b = \frac{\sum y - m \sum x}{n} = \frac{118.9 - 4.09(140)}{5}
\]
\[
= \frac{118.9 - 572.6}{5} = \frac{-453.7}{5} = -90.74
\]
5. **Final equation:**
Thus, the equation of the line of best fit is:
\[
y = 4.09x - 90.74
\]
### B. Estimate the median income for someone 32 years old
To estimate the median income for someone who is 32 years old, we can plug \( x = 32 \) into the equation of the line we found.
\[
y = 4.09(32) - 90.74
\]
\[
= 130.88 - 90.74
\]
\[
= 40.14
\]
### Conclusion
- **The equation for the line of best fit is:**
\[
y = 4.09x - 90.74
\]
- **The estimated median income for someone who is 32 years old is approximately:**
\[
\$40,140
\] (Note: Ensure consistency with units as the median income might be represented in a certain range based on context or scaling).
Answered by
GPT-4o mini
It seems there might be a misunderstanding. Let's clarify the calculation of the line of best fit step by step again to ensure we get the correct parameters.
### Given Data:
- **Age (x)**: 26, 27, 28, 29, 30
- **Median Income (y)**: 16.8, 19.1, 23.3, 25.8, 33.9
### 1. Data Points
These can be represented as:
\[
(26, 16.8), (27, 19.1), (28, 23.3), (29, 25.8), (30, 33.9)
\]
### 2. Calculating Sums Needed for the Least Squares Method
- **Number of Points (n)**: 5
- **Sum of \(x\)**:
\[
\sum x = 26 + 27 + 28 + 29 + 30 = 140
\]
- **Sum of \(y\)**:
\[
\sum y = 16.8 + 19.1 + 23.3 + 25.8 + 33.9 = 118.9
\]
- **Sum of \(xy\)**:
\[
\sum xy = (26 \cdot 16.8) + (27 \cdot 19.1) + (28 \cdot 23.3) + (29 \cdot 25.8) + (30 \cdot 33.9 = 436.8 + 515.7 + 652.4 + 748.2 + 1017 = 3370.1
\]
- **Sum of \(x^2\)**:
\[
\sum x^2 = 26^2 + 27^2 + 28^2 + 29^2 + 30^2 = 676 + 729 + 784 + 841 + 900 = 3930
\]
### 3. Calculating the Slope \(m\)
The formula for the slope \(m\) is:
\[
m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]
Plugging in our values:
\[
m = \frac{5(3370.1) - (140)(118.9)}{5(3930) - (140)^2}
\]
Calculating the numerator:
\[
= 5 \cdot 3370.1 = 16850.5
\]
\[
= 140 \cdot 118.9 = 16646
\]
Thus, the numerator becomes:
\[
16850.5 - 16646 = 204.5
\]
Calculating the denominator:
\[
= 5 \cdot 3930 = 19650
\]
\[
= (140)^2 = 19600
\]
So the denominator becomes:
\[
19650 - 19600 = 50
\]
Putting it all together for \(m\):
\[
m = \frac{204.5}{50} = 4.09
\]
### 4. Calculating the Y-Intercept \(b\)
The formula for the intercept \(b\) is:
\[
b = \frac{\sum y - m \sum x}{n}
\]
Substituting values:
\[
b = \frac{118.9 - 4.09(140)}{5}
\]
Calculating \(4.09 \cdot 140\):
\[
4.09 \cdot 140 = 572.6
\]
Now calculate \(b\):
\[
b = \frac{118.9 - 572.6}{5} = \frac{-453.7}{5} = -90.74
\]
### Final Equation of the Line
Thus, the correct equation of the line of best fit is:
\[
y = 4.09x - 90.74
\]
### 5. Estimate Median Income for Age 32
Now, substituting \(x = 32\):
\[
y = 4.09(32) - 90.74
\]
\[
= 130.88 - 90.74 = 40.14
\]
### Conclusion
- **The calculated parameters are correct:**
- **Slope (m)**: \(4.09\)
- **Y-intercept (b)**: \(-90.74\)
Thus, the equation of the line of best fit is:
\[
y = 4.09x - 90.74
\]
The estimate of the median income for someone who is 32 years old is approximately:
\[
\$40,140
\]
If you were considering the slope as \(2.5\) from a different calculation or source, it's essential to retrace those steps to understand the discrepancy. However, based on this analysis, the slope should indeed be \(4.09\).
### Given Data:
- **Age (x)**: 26, 27, 28, 29, 30
- **Median Income (y)**: 16.8, 19.1, 23.3, 25.8, 33.9
### 1. Data Points
These can be represented as:
\[
(26, 16.8), (27, 19.1), (28, 23.3), (29, 25.8), (30, 33.9)
\]
### 2. Calculating Sums Needed for the Least Squares Method
- **Number of Points (n)**: 5
- **Sum of \(x\)**:
\[
\sum x = 26 + 27 + 28 + 29 + 30 = 140
\]
- **Sum of \(y\)**:
\[
\sum y = 16.8 + 19.1 + 23.3 + 25.8 + 33.9 = 118.9
\]
- **Sum of \(xy\)**:
\[
\sum xy = (26 \cdot 16.8) + (27 \cdot 19.1) + (28 \cdot 23.3) + (29 \cdot 25.8) + (30 \cdot 33.9 = 436.8 + 515.7 + 652.4 + 748.2 + 1017 = 3370.1
\]
- **Sum of \(x^2\)**:
\[
\sum x^2 = 26^2 + 27^2 + 28^2 + 29^2 + 30^2 = 676 + 729 + 784 + 841 + 900 = 3930
\]
### 3. Calculating the Slope \(m\)
The formula for the slope \(m\) is:
\[
m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]
Plugging in our values:
\[
m = \frac{5(3370.1) - (140)(118.9)}{5(3930) - (140)^2}
\]
Calculating the numerator:
\[
= 5 \cdot 3370.1 = 16850.5
\]
\[
= 140 \cdot 118.9 = 16646
\]
Thus, the numerator becomes:
\[
16850.5 - 16646 = 204.5
\]
Calculating the denominator:
\[
= 5 \cdot 3930 = 19650
\]
\[
= (140)^2 = 19600
\]
So the denominator becomes:
\[
19650 - 19600 = 50
\]
Putting it all together for \(m\):
\[
m = \frac{204.5}{50} = 4.09
\]
### 4. Calculating the Y-Intercept \(b\)
The formula for the intercept \(b\) is:
\[
b = \frac{\sum y - m \sum x}{n}
\]
Substituting values:
\[
b = \frac{118.9 - 4.09(140)}{5}
\]
Calculating \(4.09 \cdot 140\):
\[
4.09 \cdot 140 = 572.6
\]
Now calculate \(b\):
\[
b = \frac{118.9 - 572.6}{5} = \frac{-453.7}{5} = -90.74
\]
### Final Equation of the Line
Thus, the correct equation of the line of best fit is:
\[
y = 4.09x - 90.74
\]
### 5. Estimate Median Income for Age 32
Now, substituting \(x = 32\):
\[
y = 4.09(32) - 90.74
\]
\[
= 130.88 - 90.74 = 40.14
\]
### Conclusion
- **The calculated parameters are correct:**
- **Slope (m)**: \(4.09\)
- **Y-intercept (b)**: \(-90.74\)
Thus, the equation of the line of best fit is:
\[
y = 4.09x - 90.74
\]
The estimate of the median income for someone who is 32 years old is approximately:
\[
\$40,140
\]
If you were considering the slope as \(2.5\) from a different calculation or source, it's essential to retrace those steps to understand the discrepancy. However, based on this analysis, the slope should indeed be \(4.09\).
Answered by
GPT-4o mini
Absolutely! Let's use the point-slope form to derive the equation of the line of best fit based on the provided data. The point-slope formula is given by:
\[
y - y_1 = m(x - x_1)
\]
where \(m\) is the slope of the line, and \((x_1, y_1)\) is a point on the line.
### Steps for Finding the Line of Best Fit Using Point-Slope Formula
#### Step 1: Calculate the Slope \(m\)
To calculate the slope, we can use any two points from the dataset. I'll select the first and last points for this calculation:
1. Points selected:
- \( (26, 16.8) \) and \( (30, 33.9) \)
2. Calculate the slope \(m\) using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting \( (x_1, y_1) = (26, 16.8) \) and \( (x_2, y_2) = (30, 33.9) \):
\[
m = \frac{33.9 - 16.8}{30 - 26} = \frac{17.1}{4} = 4.275
\]
#### Step 2: Use a Point and the Slope in the Point-Slope Formula
Let's use the point \( (26, 16.8) \) to write the equation:
\[
y - 16.8 = 4.275(x - 26)
\]
#### Step 3: Convert to Slope-Intercept Form (if necessary)
If needed, we can rearrange this into slope-intercept form \(y = mx + b\):
\[
y - 16.8 = 4.275(x - 26)
\]
Distributing:
\[
y - 16.8 = 4.275x - 111.15
\]
Adding \(16.8\) to both sides:
\[
y = 4.275x - 94.35
\]
### Conclusion
The equation of the line of best fit is:
\[
y = 4.275x - 94.35
\]
### Step 4: Estimate the Median Income for Age 32
Now, we can use this equation to estimate the median income for someone who is 32 years old by plugging \(x = 32\) into the equation:
\[
y = 4.275(32) - 94.35
\]
Calculating:
\[
y = 136.8 - 94.35 = 42.45
\]
### Final Estimate
The estimated median income for someone who is 32 years old is approximately:
\[
\$42,450
\]
Thank you for your patience, and I hope this approach meets your request!
\[
y - y_1 = m(x - x_1)
\]
where \(m\) is the slope of the line, and \((x_1, y_1)\) is a point on the line.
### Steps for Finding the Line of Best Fit Using Point-Slope Formula
#### Step 1: Calculate the Slope \(m\)
To calculate the slope, we can use any two points from the dataset. I'll select the first and last points for this calculation:
1. Points selected:
- \( (26, 16.8) \) and \( (30, 33.9) \)
2. Calculate the slope \(m\) using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting \( (x_1, y_1) = (26, 16.8) \) and \( (x_2, y_2) = (30, 33.9) \):
\[
m = \frac{33.9 - 16.8}{30 - 26} = \frac{17.1}{4} = 4.275
\]
#### Step 2: Use a Point and the Slope in the Point-Slope Formula
Let's use the point \( (26, 16.8) \) to write the equation:
\[
y - 16.8 = 4.275(x - 26)
\]
#### Step 3: Convert to Slope-Intercept Form (if necessary)
If needed, we can rearrange this into slope-intercept form \(y = mx + b\):
\[
y - 16.8 = 4.275(x - 26)
\]
Distributing:
\[
y - 16.8 = 4.275x - 111.15
\]
Adding \(16.8\) to both sides:
\[
y = 4.275x - 94.35
\]
### Conclusion
The equation of the line of best fit is:
\[
y = 4.275x - 94.35
\]
### Step 4: Estimate the Median Income for Age 32
Now, we can use this equation to estimate the median income for someone who is 32 years old by plugging \(x = 32\) into the equation:
\[
y = 4.275(32) - 94.35
\]
Calculating:
\[
y = 136.8 - 94.35 = 42.45
\]
### Final Estimate
The estimated median income for someone who is 32 years old is approximately:
\[
\$42,450
\]
Thank you for your patience, and I hope this approach meets your request!