The table below shows how a person's age is related to their medium income

Question

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

Answer 1

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).

Answer 2

Wouldn't a be 2.5 x

Answer 3

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 \(4.09\).

Answer 4

Use the point slope formula for a please no por favor

Answer 5

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!