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Dem bones Archaeopteryx is an extinct beast that had feathers like a bird but teeth and a long bony tail like a reptile. Only six fossil specimens are known to exist today. Because these specimens differ greatly in size, some scientists think they are different species rather than individuals from the same species. 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, using the formula on page 166. Explain how your value for r matches your graph in part (a).

Step by step solution

01

Part (a) Step 1: Given information

The following are the lengths of the femur (a leg bone) and the humerus (an upper arm bone) in centimeters:

02

Part (a) Step 2: Explanation

For scatterplot:

On horizontal $x−$axis:

Femur (in centimeters)

On vertical $y−$ axis:

Humerus (in centimeters):

Note that

All points form a linear pattern.

Thus,

We can say

All five species come from the same species.

03

Part (b) Step 1: Explanation

From the given table,

Find $xy,x^2,y^2$

Now,

Sum up every column:

$\mathrm{Î\pounds }x=291,\mathrm{Î\pounds }y=330,\mathrm{Î\pounds }xy=20040,\mathrm{Î\pounds }{x}^2=17633,\mathrm{Î\pounds }{y}^2=22790$

Formula to estimate the correlation coefficient:

$r=\frac{nâ‹\dots \mathrm{Î\pounds }xy−\mathrm{Î\pounds }xâ‹\dots \mathrm{Î\pounds }y}{\sqrt{\left[n\mathrm{Î\pounds }{x}^2-{\left(\mathrm{Î\pounds }x\right)}^2\right]}\left[n\mathrm{Î\pounds }{y}^2-{\left(\mathrm{Î\pounds }y\right)}^2\right]}$

Substitute the values:

$r=\frac{5×20040−291×330}{\sqrt{\left[5×17633−{\left(291\right)}^2\right]\left[5×22790−{\left(330\right)}^2\right]}}≈0.9941$

Because all five specimens are from the same species and have the same line of points.

Thus,

The value of correlation matches with the graph in Part (a).

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