Scenario: Linear Regression with the Palmer Penguins Data

In this project we use the Palmer Penguins dataset to practice fitting and interpreting simple linear regression models. The project has two parts. In Part 1, we work through a complete example together — making a scatterplot, computing the correlation coefficient, fitting a regression line, interpreting the slope and intercept, making a prediction, and computing a residual. In Part 2, you will carry out your own analysis from start to finish.

Part 1: Worked Example Does flipper length predict body mass in Gentoo penguins?

1A: Filtering our data

We filter to Gentoo penguins and drop any rows with missing values in our two variables of interest.

gentoo <- penguins %>%
  filter(species == "Gentoo",
         !is.na(flipper_length_mm),
         !is.na(body_mass_g))

1B: Scatterplot

We begin by making a scatterplot to look at the association before computing anything.

ggplot(gentoo, aes(x = flipper_length_mm, y = body_mass_g)) +
  geom_point(alpha = 0.6) +
  labs(
    title = "Flipper Length vs. Body Mass (Gentoo Penguins)",
    x     = "Flipper Length (mm)",
    y     = "Body Mass (g)"
  )

There is a moderate-to-strong, positive, linear association between flipper length and body mass for Gentoo penguins. Penguins with longer flippers tend to have greater body mass. There are no obvious outliers or major departures from linearity.

1C: Correlation Coefficient

cor(gentoo$flipper_length_mm, gentoo$body_mass_g, use = "complete.obs")
## [1] 0.7026665

We see that the correlation coefficient is 0.703. This positive value confirms the positive direction we saw in the scatterplot. The magnitude (0.703) indicates a moderately strong linear association — penguins with longer flippers tend to be heavier, and this tendency is fairly consistent across the sample.

1D: Fitting the Regression Line

We use lm() to fit the line of best fit, with body mass as the response variable and flipper length as the explanatory variable.

model <- lm(body_mass_g ~ flipper_length_mm, data = gentoo)
coef(model)
##       (Intercept) flipper_length_mm 
##        -6787.2806           54.6225

The regression equation is: y-hat = 54.62x - 6787.28, where x is flipper length in mm and y-hat is predicted body mass in grams.

Interpretation of the slope: For each additional 1 mm of flipper length, the model predicts body mass to increase by approximately 54.62 grams, on average.

Interpretation of the intercept: The intercept of -6787.28 grams is the predicted body mass when flipper length is 0 mm. This is physically impossible, so the intercept does not have a meaningful real-world interpretation here — it simply positions the line correctly within the range of the data.

1E: Scatterplot with Regression Line

ggplot(gentoo, aes(x = flipper_length_mm, y = body_mass_g)) +
  geom_point(alpha = 0.6) +
  geom_smooth(method = "lm", se = FALSE, color = "steelblue", linewidth = 1) +
  labs(
    title    = "Flipper Length vs. Body Mass (Gentoo Penguins)",
    x        = "Flipper Length (mm)",
    y        = "Body Mass (g)"
  )
## `geom_smooth()` using formula = 'y ~ x'

1F: Making a Prediction

What body mass does the model predict for a Gentoo penguin with a flipper length of 210 mm?

54.62*210 - 6787.28
## [1] 4682.92

The model predicts a body mass of approximately 4683 grams for a Gentoo penguin with a 210 mm flipper.

One Gentoo penguin in our dataset has a flipper length of 210 mm and an actual body mass of 4,400 grams. What is its residual?

4400 - 4683
## [1] -283

The residual is approximately -283 grams. This is negative, meaning this penguin’s actual body mass is about 283 grams below what the model predicts for a penguin of its flipper length. It sits below the regression line.

Part 2

Bill depth (mm) predicting body mass (g) for Gentoo penguins.

2A: Filter and Describe Your Data

gentoo <- penguins %>%
  filter(species == "Gentoo",
         !is.na(bill_depth_mm),
         !is.na(body_mass_g))

2B: Scatterplot

ggplot(gentoo, aes(x = bill_depth_mm, y = body_mass_g)) +
  geom_point(alpha = 0.6) +
  labs(
    title = "Bill Depth (mm) vs. Body Mass (g) (Gentoo Penguins)",
    x     = "Bill Depth (mm)",
    y     = "Body Mass (g)"
  )

Describe the association

We see a positive correlation, the strength is moderate, linearity isn’t very strong, most notable outliers are points ~(46, 5900), and ~(59, 6100).

2C: Correlation Coefficient

Compute your correlation coefficient using appropriate code. Report the value and interpret it in the context of your chosen variables.

cor(gentoo$bill_depth_mm, gentoo$body_mass_g, use = "complete.obs")
## [1] 0.719085

Through the result of the calculation for r, we find a moderately strong correlation, on a graph the data will appear to have a pull towards linearity but will remain clustered, and also contains several outliers.

2D: Fitting the Regression Line

model <- lm(body_mass_g ~ bill_depth_mm, data = gentoo)
coef(model)
##   (Intercept) bill_depth_mm 
##     -458.9852      369.4406

Regression equation:

y-hat = 369 x + (-459)

The slope indicates that for every 1 unit increase of Bill Length (mm), we observe an increase of the Body Mass (g) by 369.

When the Bill Length (mm) is 0, we observe the Body mass (g) of the penguin being -459.

Add the regression line to your scatterplot.

ggplot(gentoo, aes(x = bill_depth_mm, y = body_mass_g)) +
  geom_point(alpha = 0.6) +
  geom_smooth(method = "lm", se = FALSE, color = "steelblue", linewidth = 1) +
  labs(
    title    = "Bill Depth vs. Body Mass (Gentoo Penguins)",
    x        = "Bill Depth (mm)",
    y        = "Body Mass (g)"
  )
## `geom_smooth()` using formula = 'y ~ x'

Choose a reasonable value of x within the range of your data. Use your linear model to predict the corresponding y-hat value. Report and interpret the result.

369*14 + (-459) 
## [1] 4707

The x value I chose: 14

Predicted y: 4707

One Gentoo penguin in our dataset has a Bill Depth of 14.0 mm and an actual body mass of 4,875 grams.

4875 - 4707
## [1] 168

The residual is approximately 168 grams. This is positive, meaning this penguin’s actual body mass is about 168 grams above what the model predicts for a penguin of its bill depth. It sits above the regression line.