Chapter 3: Q.19 (page 161)

Dem bones Archaeopteryx is an extinct beast having feathers like a bird but teeth and a long bony tail like a reptile. Only six fossil specimens are known. Because these specimens differ greatly in size, some scientists think they are different species rather than individuals from the same species. We will examine some data. If the specimens belong to the same species and differ in size because some are younger than others, there should be a positive linear relationship between the lengths of a pair of bones from all individuals. An outlier from this relationship would suggest a different species. Here are data on the lengths in centimeters of the femur (a leg bone) and the humerus (a bone in the upper arm) for the five specimens that preserve both bones:
(a) Make a scatterplot. Do you think that all five specimens come from the same species? Explain.
(b) Find the correlation r step-by-step. First, find the mean and standard deviation of each variable. Then find the six standardized values for each variable. Finally, use the formula for $r$ . Explain how your value for $r$ matches your graph in (a).

Short Answer

Expert verified

a). Yes, each of the five specimens belongs to the same species.

b). The value of $r = 0.9941$ .

Step by step solution

Part (a) Step 1: Given Information

Data on the lengths in centimeters of the femur (a leg bone) and the humerus (a bone in the upper arm) for the five specimens that preserve both bones:

Part (a) Step 2: Explanation

Femur $(x)$ is on the horizontal axis, while Humerus $(y)$ is on the vertical axis.

We see that all of the points create a linear pattern (they all appear to be on the same line), implying that all five specimens belong to the same species.

Part (b) Sep 1: Given Information

Data on the lengths in centimeters of the femur (a leg bone) and the humerus (a bone in the upper arm) for the five specimens that preserve both bones:

Part (b) Step 2: Explanation

To find $X \cdot Y, X^{2}$ and $Y^{2}$ as it was done in the table below.

Part(b) Step 3: Explanation

Find the sum of every column to get:

$\sum X = 291, \sum Y = 330, \sum X \cdot Y = 20040, \sum X^{2} = 17633, \sum Y^{2} = 22790$

To calculate the correlation coefficient, use the formula below.

localid="1649931784969" $r = \frac{n \cdot \sum X Y - \sum X \cdot \sum Y}{\sqrt{[n \sum X^{2} - {(\sum X)}^{2}] \cdot [n \sum Y^{2} - {(\sum Y)}^{2}]}} = \frac{5 \cdot 20040 - 291 \cdot 330}{\sqrt{[5 \cdot 17633 - 291^{2}] \cdot [5 \cdot 22790 - 330^{2}]}} \approx 0.9941$