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	14	31	51	28	50	25
y	3	4	5	5	9	3

Complete parts (a) through (e), given Σx = 199, Σy = 29, Σx² = 7667, Σy² = 165, Σxy = 1086, and r ≈ 0.763.

(a) Draw a scatter diagram displaying the data.

(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	=
y	=
	=	+  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 r² to three decimal places. Round your answers for the percentages to one decimal place.)

r2 =
explained	%
unexplained	%

(f) For a neighborhood with x = 45 hundred jobs, how many are predicted to be entry level jobs? (Round your answer to two decimal places.)
hundred jobs

Answer 1

Part a) -

Part b) -

We calculate the following answers:

Mean of X =

Mean of y =

r =

Part c) -

X = mean of X = 33.17

Y =mean of Y = 4.83

Let the least square regression line be: y = a + b X

Formula for least square estimates of a and b are:

Hence, Y = 0.947 + 0.117 X

Part d) -

Estimated values of y are: (2.585, 4.574, 6.914, 4.223, 6.797, 3.872)

Part e) -

Coefficient of determination = r² = 0.763² = 0.582

Explained variation = 100(r² )% = 58.2%

Unexplained variation = 100(1-r² ) = 41.8%

58.2% of the total variation in y can be explained by X and the least square regression line.

41.8% of the total variation in y is unexplained by X and the least square regression line.

Part f) -

For X=45 hundred, predicted value of y = 0.947 + 0.117*45 = 6.21 hundred

Hence, for 4500 jobs the predicted entry level jobs are 621.

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