Question

Anthropologists use a linear model that relates femur length to height. The model allows an anthropologist todetermine the height of an individual when only a partial skeleton (including the femur) is found in this problem, wefind the model by analyzing the data on femur length and height for the eight males given in the tableFemur length (cm)49.948.646.345.844.343.639.738.9Height (cm)177.5174.6165.6164.7165.3164.6155.8156.71.Which scatterplot best matches the given data?A.B.2.What is the correlation coefficient for this data?

Answer 1

For the scatter plot (part 1), you should have gotten something like this.

Notice that, in general, the greater the femur length is, the taller the person is. Therefore, the relation between those two quantities has the form

`Y=mX+b,m>0`

And, the correlation coefficient is

`r=\frac{\sum ^(\square)_(\square)(x_i-\bar{x})(y_i-\bar{y})}{\sqrt[]{\sum ^(\square)_(\square)(x_i-\bar{x})^2\sum ^(\square)_(\square)(y_i-y)^2}}`

In our case,

`\begin{gathered} \bar{x}=(1)/(8)(49.9+48.6+46.3+45.8+44.3+43.6+39.7+38.9)=44.6375 \\ \bar{y}=(1)/(8)(177.5+174.6+165.6+164.7+165.3+164.6+155.8+156.7)=165.6 \end{gathered}`

Therefore,

`\begin{gathered} \sum ^(\square)_{}(x_i-\bar{x})(y_i-\bar{y}_{})=197.83 \\ \end{gathered}`

and

`\begin{gathered} \sum ^(\square)_(\square)(x_i-\bar{x})=105.9987 \\ \sum ^(\square)_(\square)(y_i-\bar{y})=399.76 \end{gathered}`

Finally,

`\Rightarrow r=\frac{197.83}{\sqrt[]{(105.9987)(399.76)}}=0.961`

Thus, the correlation coefficient is equal to 0.961