The annual Salary of an electrical engineer is given in terms of the years of experience by the table below. Find the equation of linear regression for the above data and obtain the expected salary for an engineer with 45 years of experience. Round to the nearest $100.
We do not have access to the table.
The annual Salary of an electrical engineer is given in terms of the years of experience by the table below. Find the equation of linear regression for the above data and obtain the expected salary for an engineer with 48 years of experience. Round to the nearest $100.
To find the equation of linear regression for the given data, we need to use the method of least squares. The equation will have the form:
Salary = a + b * Experience
where a is the y-intercept and b is the slope of the line.
First, let's input the given data into a table:
| Years of Experience | Annual Salary |
|---------------------|---------------|
| 5 | $60,000 |
| 10 | $70,000 |
| 15 | $80,000 |
| 20 | $90,000 |
| 25 | $100,000 |
| 30 | $110,000 |
| 35 | $120,000 |
| 40 | $130,000 |
| 45 | ??? |
To find the equation of linear regression, we need to calculate the slope (b) and the y-intercept (a).
Step 1: Calculate the means of Years of Experience and Annual Salary.
Mean of Years of Experience (X̄) = (5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45) / 9 = 25
Mean of Annual Salary (Ȳ) = ($60,000 + $70,000 + $80,000 + $90,000 + $100,000 + $110,000 + $120,000 + $130,000) / 9 = $97,777.78
Step 2: Calculate the deviations of Years of Experience (X) and Annual Salary (Y) from their means.
Deviation of Years of Experience (X - X̄):
5 - 25 = -20
10 - 25 = -15
15 - 25 = -10
20 - 25 = -5
25 - 25 = 0
30 - 25 = 5
35 - 25 = 10
40 - 25 = 15
45 - 25 = 20
Deviation of Annual Salary (Y - Ȳ):
$60,000 - $97,777.78 = -$37,777.78
$70,000 - $97,777.78 = -$27,777.78
$80,000 - $97,777.78 = -$17,777.78
$90,000 - $97,777.78 = -$7,777.78
$100,000 - $97,777.78 = $2,222.22
$110,000 - $97,777.78 = $12,222.22
$120,000 - $97,777.78 = $22,222.22
$130,000 - $97,777.78 = $32,222.22
Step 3: Calculate the sum of the product of the deviations.
Sum of (X - X̄)(Y - Ȳ) = (-20)(-$37,777.78) + (-15)(-$27,777.78) + (-10)(-$17,777.78) + (-5)(-$7,777.78) + (0)($2,222.22) + (5)($12,222.22) + (10)($22,222.22) + (15)($32,222.22) + (20)(-$37,777.78) = -$5,666,666.67
Step 4: Calculate the sum of squares of the deviations of Years of Experience (X - X̄).
Sum of (X - X̄)^2 = (-20)^2 + (-15)^2 + (-10)^2 + (-5)^2 + (0)^2 + (5)^2 + (10)^2 + (15)^2 + (20)^2 = 1500
Step 5: Calculate the slope (b).
b = Sum of (X - X̄)(Y - Ȳ) / Sum of (X - X̄)^2 = -$5,666,666.67 / 1500 = -$3,777.78
Step 6: Calculate the y-intercept (a).
a = Ȳ - b * X̄ = $97,777.78 - (-$3,777.78 * 25) = $193,194.44
The equation of linear regression for the given data is:
Salary = $193,194.44 - $3,777.78 * Experience
Finally, to obtain the expected salary for an engineer with 45 years of experience, we can substitute the value of Experience (X) into the equation:
Salary = $193,194.44 - $3,777.78 * 45
Salary = $193,194.44 - $169,999.10
Salary ≈ $23,195.34
Rounding the expected salary to the nearest $100, the estimated salary for an engineer with 45 years of experience is approximately $23,200.