Asked by Li-Yun Chang on Jun 20, 2024
Verified
Your boss would like you to estimate the fixed and variable components of a particular cost. Actual data for this cost over four recent periods appear below. Using the least-squares regression method, what is the cost formula for this cost?
A) Y = $75.89 + $1.02X
B) Y = $72.64 + $2.13X
C) Y = $0.00 + $5.04X
D) Y = $75.50 + $2.02X
Least-squares Regression
A statistical method used to determine the line of best fit by minimizing the sum of squares of the differences between observed values and the values predicted by the model.
Cost Formula
An equation used to predict the total costs associated with producing a given level of output.
- Determine financial outlays by utilizing the method of least squares regression.
- Generate cost equations for mixed expenses utilizing real-world data.
Verified Answer
SJ
Sonet JacksiJun 26, 2024
Final Answer :
D
Explanation :
Using the least-squares regression method, we can find the cost formula by calculating the slope and intercept of the line of best fit.
First, we need to calculate the variable cost per unit, which is the slope of the line. Using the formula for slope, we have:
slope = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)
where n is the number of data points, Σxy is the sum of the products of each x and y value, Σx is the sum of the x values, Σy is the sum of the y values, and Σx^2 is the sum of the squared x values.
Plugging in the values from the table above, we get:
slope = (4(6443.5) - (42)(301.65)) / (4(386) - (42)^2) = 2.02
So the variable cost per unit is $2.02.
Next, we need to calculate the fixed cost, which is the y-intercept of the line. Using the formula for the y-intercept, we have:
y-intercept = ȳ - slope(x̄)
where ȳ is the average y value, x̄ is the average x value, and slope is the variable cost per unit that we just calculated.
Plugging in the values from the table above, we get:
ȳ = (301.65 + 487.35 + 718.80 + 907.20) / 4 = 376.75
x̄ = (10 + 15 + 20 + 25) / 4 = 17.5
y-intercept = 376.75 - 2.02(17.5) = 75.50
So the fixed cost is $75.50.
Putting it all together, the cost formula is:
Y = $75.50 + $2.02X
Therefore, the best choice is D, Y = $75.50 + $2.02X.
First, we need to calculate the variable cost per unit, which is the slope of the line. Using the formula for slope, we have:
slope = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)
where n is the number of data points, Σxy is the sum of the products of each x and y value, Σx is the sum of the x values, Σy is the sum of the y values, and Σx^2 is the sum of the squared x values.
Plugging in the values from the table above, we get:
slope = (4(6443.5) - (42)(301.65)) / (4(386) - (42)^2) = 2.02
So the variable cost per unit is $2.02.
Next, we need to calculate the fixed cost, which is the y-intercept of the line. Using the formula for the y-intercept, we have:
y-intercept = ȳ - slope(x̄)
where ȳ is the average y value, x̄ is the average x value, and slope is the variable cost per unit that we just calculated.
Plugging in the values from the table above, we get:
ȳ = (301.65 + 487.35 + 718.80 + 907.20) / 4 = 376.75
x̄ = (10 + 15 + 20 + 25) / 4 = 17.5
y-intercept = 376.75 - 2.02(17.5) = 75.50
So the fixed cost is $75.50.
Putting it all together, the cost formula is:
Y = $75.50 + $2.02X
Therefore, the best choice is D, Y = $75.50 + $2.02X.
Learning Objectives
- Determine financial outlays by utilizing the method of least squares regression.
- Generate cost equations for mixed expenses utilizing real-world data.
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