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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: \(^{10}\) $$\begin{array}{lccccc}\hline \text { Femur }(x): & 38 & 56 & 59 & 64 & 74 \\\\\text { Humerus }(y): & 41 & 63 & 70 & 72 & 84 \\\\\hline\end{array}$$ (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 154 . Explain how your value for \(r\) matches your graph in part (a).

Step by step solution

01

Plot the Scatterplot

To make a scatterplot, plot the femur lengths on the x-axis and the humerus lengths on the y-axis. The data points are (38, 41), (56, 63), (59, 70), (64, 72), and (74, 84). Plot these points on a coordinate plane to form a visual representation of the relationship between the two bones' lengths.

02

Analyze the Scatterplot

Examine the scatterplot to assess whether there is a pattern that suggests a linear relationship. If the points roughly line up in a straight line, this indicates a potential positive linear relationship. Review how closely the points adhere to an imaginary straight line to determine if they could belong to the same species.

03

Calculate Means

To find the correlation, first calculate the mean of both the femur and humerus lengths. \[\bar{x} = \frac{38 + 56 + 59 + 64 + 74}{5} = 58.2\]\[\bar{y} = \frac{41 + 63 + 70 + 72 + 84}{5} = 66\]

04

Calculate Differences from Means

Compute the difference between each data point and their respective mean. Use these differences in further calculations. For femur lengths: \[ x_i - \bar{x} = 38 - 58.2, 56 - 58.2, 59 - 58.2, 64 - 58.2, 74 - 58.2 \]For humerus lengths: \[ y_i - \bar{y} = 41 - 66, 63 - 66, 70 - 66, 72 - 66, 84 - 66 \]

05

Compute Products of Differences

Calculate the product of the differences from the means for each data point pair. For example, for the first data pair:\[(x_1 - \bar{x})(y_1 - \bar{y}) = (38 - 58.2)(41 - 66) = 20.2(20) \]Continue this calculation for each pair.

06

Sum of Products

Add together all the products calculated in Step 5.

07

Calculate Sum of Squares

Determine the sum of squares for both femur and humerus lengths. \[\sum (x_i - \bar{x})^2 \]\[\sum (y_i - \bar{y})^2 \]

08

Compute Correlation Coefficient (r)

Finally, use the correlation formula:\[r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}\]Plug in the values computed from previous steps to find the correlation coefficient.

09

Interpret the Correlation and Graph

Compare the scatterplot with the calculated correlation coefficient. A correlation value of close to 1 implies a strong linear relationship, supporting the view that differences in size are due to age, implying the specimens are from the same species.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatterplot

In the study of paleontology, understanding relationships between different biological measurements is crucial. A scatterplot is an essential tool for visualizing these relationships. Specifically, a scatterplot displays data points on a coordinate plane, helping reveal patterns or trends between two variables. In the context of studying Archaeopteryx fossils, we plot the lengths of the femur and humerus bones to see if they follow a pattern.

To create a scatterplot, plot each data pair, with femur lengths on the x-axis and humerus lengths on the y-axis. Connect these points visually, but do not draw lines between them. The scatterplot allows us to identify any visual pattern that might suggest a linear relationship or highlight outliers, which are points that deviate significantly from others.

The goal here is to observe if the data points form a line that can indicate a positive linear trend. This trend would support the hypothesis that size differences result from age rather than species variation. If the points diverge significantly from a straight line, it may hint at different species within the sample.

Correlation Coefficient

The correlation coefficient, denoted by \(r\), is a statistical measure that expresses the extent of a linear relationship between two variables. It quantifies how closely the data in a scatterplot line up along a straight line. In paleontology, as with our Archaeopteryx data, it helps confirm or refute the hypothesis about species or age differences.

The calculation of \(r\) involves several steps: computing the means of both datasets, determining the deviations of each data point from these means, and using these deviations to find products and sums necessary for the formula. **Key steps include:**

• Calculate the means, \(\bar{x}\) and \(\bar{y}\), for both sets of data.
• Compute deviations \((x_i - \bar{x})\) and \((y_i - \bar{y})\).
• Determine the sum of the products of these deviations.
• Find the sum of squares for each dataset.

Then, plug these into the correlation coefficient formula: \[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \]
A correlation coefficient \(r\) close to 1 suggests a strong positive linear relationship, indicating that differences among the fossils may be attributed to age rather than separate species.

Linear Relationship

A linear relationship in statistics implies that the change in one variable is related to a change in another variable in a proportional manner. For archaeopteryx fossils, a linear relationship between femur and humerus lengths would suggest that the differences in sizes are age-related. It is one of the key indicators of trying to determine if variations in specimens are due to developmental stages rather than different species.

If we plot these bone lengths and the points form a straight line, it implies a consistent rate of growth between the femur and humerus bones as the creature ages. However, deviations from a linear pattern could suggest other variables at play, such as different species.

In our analysis, establishing a linear relationship assists researchers in arguing against separate species hypotheses when supporting evidence is needed to lean towards age-related variations. It's a powerful tool for forming conclusions based on visual and statistical data analysis, providing significant insights into paleontological studies.