A company determines an employee's starting salary according to the number of years of experience, as detailed in the table.
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 linear regression. The equation of the line of best fit is given by y = mx + b, where y is the dependent variable (salary), x is the independent variable (years of experience), m is the slope of the line, and b is the y-intercept.
Using the formula for m (slope) and b (y-intercept) in linear regression:
m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)
b = (Σy - mΣx) / n
Where n is the number of data points, Σxy is the sum of the products of the x and y values, Σx is the sum of the x values, Σy is the sum of the y values, and Σx^2 is the sum of the squares of the x values.
In this case, we have the following data:
x: 0, 1, 2, 3, 4, 5
y: 40000, 42150, 44260, 46785, 48820, 51126
First, calculate the sums:
Σx = 0 + 1 + 2 + 3 + 4 + 5 = 15
Σy = 40000 + 42150 + 44260 + 46785 + 48820 + 51126 = 273141
Σxy = (0 * 40000) + (1 * 42150) + (2 * 44260) + (3 * 46785) + (4 * 48820) + (5 * 51126) = 1323655
Σx^2 = (0^2) + (1^2) + (2^2) + (3^2) + (4^2) + (5^2) = 55
Using the formulas for m and b, we can calculate:
m = (6 * 1323655 - 15 * 273141) / (6 * 55 - 15^2) = 65515 / 45 ≈ 1455.89
b = (273141 - 1455.89 * 15) / 6 = 1316.84
So the equation for the line of best fit is y = 1455.89x + 1316.84
Plugging in x = 7:
y = 1455.89(7) + 1316.84
y ≈ 10205.23 + 1316.84
y ≈ 11522.07
Rounding to the nearest dollar, the predicted salary for an employee with 7 years of experience is approximately $11,522. Therefore, the correct answer is not among the provided options.
To predict the salary for an employee with 7 years of experience, we will use the equation for the line of best fit.
The equation for a line in slope-intercept form is y = mx + b, where y is the dependent variable (salary), x is the independent variable (years of experience), m is the slope of the line, and b is the y-intercept.
To find the equation for the line of best fit, we need to calculate the slope (m) and y-intercept (b) using the given data points.
First, let's calculate the slope (m) using the formula:
m = (Σxy - (Σx)(Σy)/n) / (Σx^2 - (Σx)^2)/n)
Where Σxy is the sum of the products of years of experience (x) and salary (y), Σx is the sum of years of experience, Σy is the sum of salaries, Σx^2 is the sum of squares of years of experience and n is the number of data points.
Let's calculate the values needed for the equation:
Σxy = (0*40000) + (1*42150) + (2*44260) + (3*46785) + (4*48820) + (5*51126)
= 0 + 42150 + 88520 + 140355 + 195280 + 255630
= 771355
Σx = 0 + 1 + 2 + 3 + 4 + 5
= 15
Σy = 40000 + 42150 + 44260 + 46785 + 48820 + 51126
= 273141
Σx^2 = (0^2) + (1^2) + (2^2) + (3^2) + (4^2) + (5^2)
= 0 + 1 + 4 + 9 + 16 + 25
= 55
n = 6 (since there are 6 data points)
Now, let's substitute these values into the slope formula:
m = (771355 - (15 * 273141)/6) / (55 - (15)^2)/6)
Simplifying,
m = (771355 - 409711)/(-310)
m = 361644 / (-310)
m ≈ -1164.99
Next, let's calculate the y-intercept (b) using the formula:
b = Σy/n - m(Σx/n)
Substituting the values,
b = (273141/6) - (-1164.99 * 15)/6
b = 45523.5 + 2924.985
b ≈ 48448.49
Now that we have the values for m and b, we can substitute them into the equation y = mx + b to predict the salary for an employee with 7 years of experience.
y = (-1164.99 * 7) + 48448.49
y ≈ -8154.93 + 48448.49
y ≈ $40293.56
Rounding to the nearest dollar, the predicted salary for an employee with 7 years of experience is approximately $40,294.
None of the provided answers are an exact match, so the closest answer is a. $52,900.