Question

​​​​​​​The file penguins.csv contains measurements for penguins foraging near Palmer Station in Antarctica. The dataset includes bill (or beak) length (in mm) and bill depth (in mm) for 333 penguins from three species (Adelie, Chinstrap and Gentoo). In the following context, treat bill length as the explanatory variable and bill depth as response variable.

a) Perform a linear regression for predicting penguin bill depth from bill length. Print out the slope and intercept. Interpret the intercept in the context of the problem.
b) Compute the coefficient of determination (2) for the regression above. Interpret it.
c) Obtain the least-squared regression line (by printing out the slope and intercept) for predicting a penguin’s bill depth from bill length for only Adelie penguins. Interpret the slope in the context of the problem.
d) Repeat the steps (regression and interpretation of slope) in part c) for the other two species.

Answer 1

Linear regression is used to find a best-fit line by minimizing the residuals, with the slope indicating the rate of change of the dependent variable and the y-intercept often being non-practical. The coefficient of determination (R²) measures how well the regression line fits the data, and separate regression lines calculated for each species of penguins highlight differences in species-specific characteristics.

Step-by-step explanation:

The process of linear regression involves finding the best-fit line through a set of data points by minimizing the sum of the squares of the vertical distances (residuals) of the points from the line. In a regression equation of the form ý = a + bx, 'b' represents the slope of the line and 'a' represents the y-intercept. The slope indicates the rate at which the dependent variable (in this case, bill depth) changes for each unit change in the independent variable (bill length). The y-intercept is the value of the dependent variable when the independent variable is zero, which often does not have a practical interpretation when extrapolated beyond the range of observed data.

To interpret the coefficient of determination, denoted as R², it represents the proportion of variability in the dependent variable that is explained by the linear relationship with the independent variable. The closer R² is to 1, the better the fit of the regression line to the data. In the context of penguin bill measurements, calculating R² would tell us how much of the variation in bill depth can be predicted from bill length.

For each species of penguin, a separate least-squares regression line is calculated to predict bill depth from bill length. The slope of each line for each species will indicate how bill depth changes with bill length specifically for Adelie, Chinstrap, and Gentoo penguins, which may differ due to species-specific characteristics.