A company determines an employee's starting salary according to the number of years of experience, as detailed in the table.
this is supposed to be a chart
Years of experience: 0, 1, 2, 3, 4, 5 Salary: $40000, $42150, $44260, $46785, $48820, $51126
Use the equation for the line of best fit to predict the salary for an employee with 7 years of experience? (Round your answer to the nearest dollar.)
a.$52,900
b.$53,340
c.$53,914
d.$55,573
use one of the provided answers
To find the equation for the line of best fit, we can use the given data to calculate the slope and y-intercept.
The slope is calculated using the formula:
slope = (Σ(xy) - (Σx)(Σy) / n(Σx^2) - (Σx)^2)
Σ(xy) = (0)(40000) + (1)(42150) + (2)(44260) + (3)(46785) + (4)(48820) + (5)(51126) = 100,4820
Σx = 0 + 1 + 2 + 3 + 4 + 5 = 15
Σy = 40000 + 42150 + 44260 + 46785 + 48820 + 51126 = 273,141
Σx^2 = (0^2) + (1^2) + (2^2) + (3^2) + (4^2) + (5^2) = 55
slope = (100,4820 - (15)(273,141) / (6)(55) - (15)^2)
slope = (100,4820 - 4,096,415) / (330 - 225)
slope = -3,995,595 / 105
slope ≈ -38,052.33
The y-intercept can be calculated using the formula:
y-intercept = Σy / n - slope(Σx) / n
y-intercept = 273,141 / 6 - (-38,052.33)(15) / 6
y-intercept = 45,523.5 + 9,027.12
y-intercept ≈ 54,550.62
So, the equation for the line of best fit is y = -38,052.33x + 54,550.62.
To predict the salary for an employee with 7 years of experience (x = 7), we substitute x = 7 into the equation:
y = -38,052.33(7) + 54,550.62
y ≈ -266,366.31 + 54,550.62
y ≈ 288,916.93
Rounded to the nearest dollar, the predicted salary for an employee with 7 years of experience is $289,000.
Therefore, none of the provided answers (a, b, c, or d) are correct.
To find the line of best fit equation, we first need to calculate the slope and y-intercept.
For the given data, let's consider the years of experience (x) as the independent variable and the salary (y) as the dependent variable.
Using the formula for the slope (m):
m = (Σ(x*y) - (Σx * Σy) / n) / (Σ(x^2) - (Σx)^2 / n)
where:
Σ represents the sum
Σ(x) represents the sum of the independent variable (years of experience)
Σ(y) represents the sum of the dependent variable (salary)
x^2 represents the sum of the squares of the independent variable
n represents the number of data points
Let's calculate the values needed for the slope:
Σ(x) = 0 + 1 + 2 + 3 + 4 + 5 = 15
Σ(y) = 40000 + 42150 + 44260 + 46785 + 48820 + 51126 = 273141
Σ(x*y) = (0*40000) + (1*42150) + (2*44260) + (3*46785) + (4*48820) + (5*51126) = 1054110
Σ(x^2) = (0^2) + (1^2) + (2^2) + (3^2) + (4^2) + (5^2) = 55
n = 6
Substituting these values into the formula for the slope, we have:
m = (1054110 - (15*273141) / 6) / (55 - (15^2) / 6)
m = (1054110 - 4097115) / (55 - 225 / 6)
m = (-3043005) / (55 - 37.5)
m = (-3043005) / (17.5)
m ≈ -173942.86
Next, let's calculate the y-intercept (b) using the formula:
b = (Σy - m * Σx) / n
b = (273141 - (-173942.86 * 15)) / 6
b = (273141 + 2609142.9) / 6
b ≈ 461526.98
Now we have the slope (m ≈ -173942.86) and y-intercept (b ≈ 461526.98).
Using the equation for the line of best fit:
y = mx + b
we can substitute the value of x (7 years of experience) to predict the salary:
y = -173942.86 * 7 + 461526.98
y ≈ -1217600.02 + 461526.98
y ≈ 339126.96
Rounding the answer to the nearest dollar, we get the predicted salary for an employee with 7 years of experience as approximately $339,127.
None of the provided answers match this result, so please double-check your options or calculations.