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 15 32 51 28 50 25y 3 3 7 5 9 3Complete parts (a) through (e), given Σx = 201, Σy = 30, Σx2 = 7759, Σy2 = 182, Σxy = 1163, and r ≈ 0.872.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.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.)d. Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line.

Answer 1

Answer:

The sample correlation coefficient is, r = 0.8722.

The equation of the least-squares line is:

$y= -0.161+0.154x$

Step-by-step explanation:

(a)

The scatter diagram displaying the data for X : total number of jobs in a given neighborhood and Y : number of entry-level jobs in the same neighborhood is shown below.

(b)

The table attached below verifies the values of $\sum X,\ \sum Y,\ \sum X^{2},\ \sum Y^{2}\ \text{and}\ \sum XY$.

The sample correlation coefficient is:

$\begin{aligned}r~&=~\frac{n\cdot\sum{XY} - \sum{X}\cdot\sum{Y}} {\sqrt{\left[n \sum{X^2}-\left(\sum{X}\right)^2\right] \cdot \left[n \sum{Y^2}-\left(\sum{Y}\right)^2\right]}} \\r~&=~\frac{ 6 \cdot 1163 - 201 \cdot 30 } {\sqrt{\left[ 6 \cdot 7759 - 201^2 \right] \cdot \left[ 6 \cdot 182 - 30^2 \right] }} \approx 0.8722\end{aligned}$

Thus, the sample correlation coefficient is, r = 0.8722.

(c)

The slope and intercept are:

$\begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 30 \cdot 7759 - 201 \cdot 1163}{ 6 \cdot 7759 - 201^2} \approx -0.161 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 6 \cdot 1163 - 201 \cdot 30 }{ 6 \cdot 7759 - \left( 201 \right)^2} \approx 0.154\end{aligned}$

The equation of the least-squares line is:

$y= -0.161+0.154x$

(d)

The least-squares line is graphed in the diagram below.