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 16 32 51 28 50 25

y 1 3 7 5 9 3

Complete parts (a) through (e), given Σx = 202, Σy = 28, Σx2 = 7790, Σy2 = 174, Σxy = 1134, and r ≈ 0.924.

(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 y hat = 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 =

y hat = + 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 = 31 hundred jobs, how many are predicted to be entry level jobs? (Round your answer to two decimal places.)_________ hundred jobs

Answer 1

(a)

The scatter diagram is,

(b)

The given sums and the value of the sample correlation coefficient r are,

Σx = 202

Σy = 28

Σx² = 7790

Σy² = 174

Σxy = 1134

r = 0.924.

(c)

b = SSxy / SSxx = 191.3333 / 989.3333 = 0.193

a = = 4.67 - 0.193 * 33.67 = -1.828

y hat = 1.828 + 0.193 x

(d)

The least-squares line is,

(e)

The coefficient of determination r² = 0.924² = 0.854

Percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line = 85.4%

r2 = 0.854

explained_= 85.4_____- %

unexplained = 100 - 85.4 = 14.6_____%

(f)

For x = 31,

Predicted entry level jobs =  -1.828 + 0.193 * 31 = 4.16 hundred jobs