A sociologist is interested in the relations between x = number of job changes and y = annual salary (in thousands of dollars) for people living in the Nashville area. A random sample of 10 people employed in Nashville provided the following information.
x (Number of job changes) 4 7 5 6 1 5 9 10 10 3
y (Salary in $1000) 33 37 34 32 32 38 43 37 40 33
Complete parts (a) through (c), given ‡”x = 60, ‡”y = 359, ‡”x2 = 442, ‡”y2 = 226, 125 and ‡”yz = 2231.
(d) If someone had x = 2 job changes, what does the least-squares line predict for y, the annual salary?
To answer this question, we can use the least-squares regression line. The equation of the least-squares regression line is given by:
y = a + bx
where:
- y is the dependent variable (annual salary)
- x is the independent variable (number of job changes)
- a is the y-intercept of the line
- b is the slope of the line
To find the values of a and b, we need to use the following equations:
a = (‡”y * ‡”x2 - ‡”x * ‡”yz) / (n * ‡”x2 - ‡”x^2)
b = (n * ‡”yz - ‡”y * ‡”x) / (n * ‡”x2 - ‡”x^2)
where:
- ‡”y is the sum of the dependent variable values minus the mean of y
- ‡”x is the sum of the independent variable values minus the mean of x
- ‡”x2 is the sum of the squares of the independent variable values minus the mean of x squared
- ‡”y2 is the sum of the squares of the dependent variable values minus the mean of y squared
- ‡”yz is the sum of the product of the independent variable values minus the mean of x and the dependent variable values minus the mean of y
- n is the number of observations
Let's calculate the values step by step:
(a) Calculate ‡”y and ‡”x:
‡”y = sum of y - (n * mean of y) = 359 - (10 * (359 / 10)) = 359 - 359 = 0
‡”x = sum of x - (n * mean of x) = 60 - (10 * (60 / 10)) = 60 - 60 = 0
(b) Calculate ‡”x^2, ‡”y2, and ‡”yz:
‡”x^2 = sum of x^2 - (n * mean of x squared) = 442 - (10 * ((60 * 60) / 10)) = 442 - (10 * 360) = 442 - 3600 = -3158
‡”y2 = sum of y^2 - (n * mean of y squared) = 226,125 - (10 * ((359 * 359) / 10)) = 226,125 - (10 * 128,881.1) = 226,125 - 1,288,811 = -1,062,686
‡”yz = sum of (x * y) - (n * mean of x * mean of y) = 2231 - (10 * (0 * (359 / 10))) = 2231
(c) Calculate the values of a and b:
a = (‡”y * ‡”x2 - ‡”x * ‡”yz) / (n * ‡”x2 - ‡”x^2) = (0 * (-3158) - 0 * 2231) / (10 * (-3158) - 0) = 0 / -31580 = 0
b = (n * ‡”yz - ‡”y * ‡”x) / (n * ‡”x2 - ‡”x^2) = (10 * 2231 - 0 * 0) / (10 * (-3158) - 0) = 22,310 / -31,580 = -0.706
Therefore, the equation of the least-squares regression line is:
y = 0 + (-0.706x) = -0.706x
(d) To predict the annual salary (y) for someone with x = 2 job changes, we can substitute x = 2 into the equation of the least-squares regression line:
y = -0.706 * 2 = -1.412
So, the least-squares line predicts an annual salary of approximately -1.412 (thousand dollars) for someone with 2 job changes.