In: Statistics and Probability
A sociologist is interested in the relation 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) | 6 | 4 | 5 | 6 | 1 | 5 | 9 | 10 | 10 | 3 |
y (Salary in $1000) | 36 | 32 | 35 | 32 | 32 | 38 | 43 | 37 | 40 | 33 |
Σx = 59; Σy = 358; Σx2 = 429; Σy2 = 12,944; Σxy = 2,189
(a) Find x, y, b, and the equation of the least-squares line. (Round your answers for x and y to two decimal places. Round your least-squares estimates to three decimal places.)
x | = | |||||
y | = | |||||
b | = | |||||
+ x (c) Find the sample correlation coefficient r and the coefficient of determination. (Round your answers to three decimal places.)
What percentage of variation in y is explained by the least-squares model? (Round your answer to one decimal place.) % (d) If someone had x = 3 job changes, what does the least-squares line predict for y, the annual salary? (Round your answer to two decimal places.) thousand dollars |
Part a)
X | Y | X * Y | X2 | Y2 | |
6 | 36 | 216 | 36 | 1296 | |
4 | 32 | 128 | 16 | 1024 | |
5 | 35 | 175 | 25 | 1225 | |
6 | 32 | 192 | 36 | 1024 | |
1 | 32 | 32 | 1 | 1024 | |
5 | 38 | 190 | 25 | 1444 | |
9 | 43 | 387 | 81 | 1849 | |
10 | 37 | 370 | 100 | 1369 | |
10 | 40 | 400 | 100 | 1600 | |
3 | 33 | 99 | 9 | 1089 | |
Total | 59 | 358 | 2189 | 429 | 12944 |
X̅ = Σ( Xi / n ) = 59/10 = 5.9
Y̅ = Σ( Yi / n ) = 358/10 = 35.8
Equation of regression line is Ŷ = a + bX
b = 0.949
a =( Σ Y - ( b * Σ X) ) / n
a =( 358 - ( 0.9493 * 59 ) ) / 10
a = 30.199
Equation of regression line becomes Ŷ = 30.199 + 0.949 X
Part c)
r = 0.756
Coefficient of Determination
R2 = r2 = 0.571
Explained variation = 0.571* 100 = 57.1%
Unexplained variation = 1 - 0.571* 100 = 42.9%
Part d)
When X = 3
Ŷ = 30.199 + 0.949 X
Ŷ = 30.199 + ( 0.949 * 3 )
Ŷ = 33.05