Question

In: Statistics and Probability

A sociologist is interested in the relation between x = number of job changes and y...

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.)

r =
r2 =   

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

Solutions

Expert Solution

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


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