Question

In each of Exercises 4.169 and 4.170,

a. construct a scatterplot for the data.

b. decide whether using the rank correlation coefficient is reasonable.

c. decide whether using the linear correlation coefficient is reasonable.

d. find and interpret the rank correlation coefficient.
(Use technology) The data from Exercise 4.80 for volume, in cubic feet, and diameter at breast height, in inches, for$70$ shortleaf pines are on the WeissStats site.

Short answer

The answers for the subparts have been given below.

Step-by-step solution

Step 1. Given Information

For volume, in cubic feet, and diameter at breast height, in inches, $70$shortleaf pines are on the WeissStats site.

Step 2. The answer for (a)

For constructing the scatterplot on MINITAB follow these steps:

1. Choose Graph>Scatterplot
2. Choose WithConnect Line,and then clickOK.
3. UnderY Variables, enter a column of Volume.
4. UnderX Variables, enter a column of Diameter.
5. Click OK.

Step 3. Continuation of answer (a)

The MINITAB output:

After observing the graph we can say that there is a positive linear relationship between Diameter and Volume.

Step 4. The answer for (b)

To decide whether using the rank correlation coefficient is reasonable or not.

The rank correlation coefficient is reasonable because the variable diameter increases as the variable volume also increase. In other words, the smaller values of diameter are related to the smaller values of volume, and the larger values of diameter are related to the larger values of volume.

Step 5. The answer for (c)

To decide whether using the linear correlation coefficient is reasonable or not.

Using the linear correlation coefficient is not reasonable because there is a convex upward curved pattern. Moreover, the variable diameter increases as the variable volume also increase. Therefore, the given data does not seem to be a linear pattern.

Step 6. The answer for (d)

Determine the rank correlation coefficient by MINITAB.

First find the rank for diameter and volume by using MINITAB.

MINITAB procedure:

1. Select Data>Rank.
2. In Rank data in: Select Diameter.
3. In Store Ranks in: choose Rankdiameter.
4. In Rank data in: Select Volume.
5. In Store Ranks in: choose Rankvolume.

Step 7. Continuation of the answer (d)

Correlation:

MINITAB procedure:

1. Select Stat >Basic Statistics > Correlation.
2. In Variables, select Rankdiameter and Rankvolume from the box on the left.
3. Click OK.

MINITAB output:

Correlation: Rank Diameter, Rank Volume.

Pearson correlation of Rank Time and Rank Score $=0.988$

$P-$value$=0.000$

From the MINITAB output, the rank correlation coefficient is $0.988$.

Interpretation:

The value of the rank correlation coefficient is $0.988$. It indicates that there is a strong positive linear relationship between the variables rankdiameter and rankvolume.