Question

In exercise 10-1 12. Clusters Refer to the following Minitab-generated scatterplot. The four points in the lower left corner are measurements from women, and the four points in the upper right corner are from men.

a. Examine the pattern of the four points in the lower left corner (from women) only, and subjectively determine whether there appears to be a correlation between x and y for women.

b. Examine the pattern of the four points in the upper right corner (from men) only, and subjectively determine whether there appears to be a correlation between x and y for men.

c. Find the linear correlation coefficient using only the four points in the lower left corner (for women). Will the four points in the upper left corner (for men) have the same linear correlation coefficient?

d. Find the value of the linear correlation coefficient using all eight points. What does that value suggest about the relationship between x and y?

e. Based on the preceding results, what do you conclude? Should the data from women and the data from men be considered together, or do they appear to represent two different and distinct populations that should be analyzed separately?

Short answer

a.There seems to be no linear correlation among x and y for women.

b. There seems to be no linear correlation among x and y for men.

c. The correlation coefficient between x and y for women is 0. The value of the correlation coefficient between x and y is the same for men.

d.The correlation between x and y when all the 8 points are taken into account is equal to 0.985.

e. The points corresponding to women and men appear to represent two different and distinct populations and should be analyzed separately.

Step-by-step solution

Given Information

A scatterplot is constructed for the pairs of measurements for women and men.

Examination of  linear correlation

a.

Since the points on the lower-left corner do not seem to follow an increasing straight-line trend or a decreasing straight-line trend, there seems to be no linear correlation among x and y for women.

b.

Similarly, the points on the upper right corner neither follow an increasing straight-line trend or a decreasing straight-line trend, there seems to be no linear correlation among x and y for men.

Computation of correlation coefficient

c.

To calculate the correlation coefficient between x and y for women, the following computations are made:

x	y	$x^2$	$y^2$	$xy$
1	1	1	1	1
1	2	1	4	2
2	1	4	1	2
2	2	4	4	4
$\sum x=6$$$	$\sum y=6$	$\sum x^2=10$	$\sum y^2=10$	$\sum xy=9$

The formula to calculate correlation coefficient r is given by

$\begin{array}{rl}r& =\frac{n\sum xy-\left(\sum x\right)\left(\sum y\right)}{\sqrt{n\left(\sum x^2\right)-{\left(\sum x\right)}^2}\sqrt{n\left(\sum y^2\right)-{\left(\sum y\right)}^2}}\\ & =\frac{4\left(9\right)-\left(6\right)\left(6\right)}{\sqrt{4\left(10\right)-{\left(6\right)}^2}\sqrt{4\left(10\right)-{\left(6\right)}^2}}\\ & =0\end{array}$

Thus, the correlation coefficient between x and y for women is 0.

To calculate the correlation coefficient between x and y for men, consider the following calculations:

x	y	$x^2$	$y^2$	$xy$
9	9	81	81	81
9	10	81	100	90
10	9	100	81	90
10	10	100	100	100
$\sum x=38$$$	$\sum y=38$	$\sum x^2=362$	$\sum y^2=362$	$\sum xy=361$

The formula to calculate correlation coefficient r is given by

$\begin{array}{rl}r& =\frac{n\sum xy-\left(\sum x\right)\left(\sum y\right)}{\sqrt{n\left(\sum x^2\right)-{\left(\sum x\right)}^2}\sqrt{n\left(\sum y^2\right)-{\left(\sum y\right)}^2}}\\ & =\frac{4\left(361\right)-\left(38\right)\left(38\right)}{\sqrt{4\left(362\right)-{\left(38\right)}^2}\sqrt{4\left(362\right)-{\left(38\right)}^2}}\\ & =0\end{array}$

Thus, the correlation coefficient between x and y for men is 0.

Computation of the correlation coefficient with all 8 points

d.

To compute the correlation between all the 8 points, consider the following calculations:

x	Y	$x^2$	$y^2$	$xy$
1	1	1	1	1
1	2	1	4	2
2	1	4	1	2
2	2	4	4	4
9	9	81	81	81
9	10	81	100	90
107	9	100	81	90
10	10	100	100	100
$\sum x=44$$$	$\sum y=44$	$\sum x^2=372$	$\sum y^2=372$	$\sum xy=370$

The correlation coefficient is equal to:

$\begin{array}{rl}r& =\frac{n\sum xy-\left(\sum x\right)\left(\sum y\right)}{\sqrt{n\left(\sum x^2\right)-{\left(\sum x\right)}^2}\sqrt{n\left(\sum y^2\right)-{\left(\sum y\right)}^2}}\\ & =\frac{8\left(370\right)-\left(44\right)\left(44\right)}{\sqrt{8\left(372\right)-{\left(44\right)}^2}\sqrt{8\left(372\right)-{\left(44\right)}^2}}\\ & =0.985\end{array}$

Thus, the correlation between x and y when all the 8 points are taken into account is equal to 0.985.

Analysis of the correlation

e.

All of the points corresponding to women lie on the lower-left corner while all of the points corresponding to men lie on the upper-right corner.

There seems to be no association/mix-up of the two sets of values.

Therefore, the points corresponding to women and menappear to represent two different and distinct populations and should be analyzed separately.