Tail-feather length in birds is sometimes a sexually dimorphic trait; that is, the trait differs substantially for males and for females. Researchers studied the relationship between weight (x) and tail-feather length (y) in a sample of 5 wild male long-tailed finches. Here are the data:
$$\begin{array} { c c }
\text { weight } ( \mathrm { g } ) & \text { tail length } ( \mathrm { mm } ) \
20.8 & 82.5 \
19.1 & 82.5 \
15.9 & 67.0 \
16.7 & 70.5 \
15.7 & 73.5
\end{array}$$ Using appropriate software, what is the numerical value of the y intercept for the least-squares regression line?

Question

Tail-feather length in birds is sometimes a sexually dimorphic trait; that is, the trait differs substantially for males and for females. Researchers studied the relationship between weight (x) and tail-feather length (y) in a sample of 5 wild male long-tailed finches. Here are the data:
$$\begin{array} { c c } 
\text { weight } ( \mathrm { g } ) & \text { tail length } ( \mathrm { mm } ) \
20.8 & 82.5 \
19.1 & 82.5 \
15.9 & 67.0 \
16.7 & 70.5 \
15.7 & 73.5
\end{array}$$ Using appropriate software, what is the numerical value of the y intercept for the least-squares regression line?

Quizplus · Accepted Answer

The y-intercept of the least-squares regression line represents the value of $y$ (tail length) when $x$ (weight) is 0. Given the nature of the data and the relationship between weight and tail length, calculating the exact y-intercept requires performing a regression analysis, which involves using statistical software or a calculator to fit a line that minimizes the sum of the squared differences between the observed values and the values predicted by the line. The correct answer, 25.547, is obtained from such an analysis, indicating that if the model were to be extrapolated back to a bird weighing 0 grams (which is not practical in real terms), the model predicts a tail length of 25.547 mm. This value is specific to the data provided and the calculation method used.

Tail-Feather Length in Birds Is Sometimes a Sexually Dimorphic Trait;