Question

An economist is studying the job market in Denver area neighborhoods. Let x represent the total number of jobs in a given neighborhood, and let y represent the number of entry-level jobs in the same neighborhood. A sample of six Denver neighborhoods gave the following information (units in hundreds of jobs).

x	17	33	52	28	50	25
y	3	3	6	5	9	3

Complete parts (b) through (e), given Σx = 205, Σy = 29, Σx² = 7991, Σy² = 169, Σxy = 1127, and r ≈ 0.807.

(b) Verify the given sums Σx, Σy, Σx², Σy², Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.)

Σx =
Σy =
Σx2 =
Σy2 =
Σxy =
r =

(c) Find x, and y. Then find the equation of the least-squares line  = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)

x-bar	=
y-bar	=
	=	____ + ____ x

(d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line.

(e) Find the value of the coefficient of determination r². What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r²to three decimal places. Round your answers for the percentages to one decimal place.)

r2 =
explained	%
unexplained	%

(f) For a neighborhood with x = 31 hundred jobs, how many are predicted to be entry level jobs? (Round your answer to two decimal places.)
____ hundred jobs

Answer 1

Part b)

ΣX = 205
ΣY = 29
ΣX * Y = 1127
ΣX2 = 7991
ΣY2 = 169

r = 0.807

Part c)

X̅ = Σ( Xi / n ) = 205/6 = 34.17
Y̅ = Σ( Yi / n ) = 29/6 = 4.83

Equation of regression line is Ŷ = a + bX


b = 0.138
a =( Σ Y - ( b * Σ X) ) / n
a =( 29 - ( 0.138 * 205 ) ) / 6
a = 0.119
Equation of regression line becomes Ŷ = 0.119 + 0.138 X

part d )

Part e)

Coefficient of Determination
= 0.652
Explained variation = 0.652* 100 = 65.2%
Unexplained variation = 1 - 0.652* 100 = 34.8%

Part f)

When X = 31
Ŷ = 0.119 + 0.138 X
Ŷ = 0.119 + ( 0.138 * 31 )
Ŷ = 4.4