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: Your Turn

Now it is your turn to carry out a complete regression analysis. Choose one of the following options.

Option 1: Bill depth (mm) predicting body mass (g) for Gentoo penguins.

Option 2: Bill length (mm) predicting body mass (g) for Chinstrap penguins.

Option 3: Bill length (mm) predicting bill depth (mm) for Chinstrap penguins.

State which option you chose:

Option 3

2A: Filter and Describe Your Data

Filter to the appropriate species and drop rows with missing values in your two variables.

chinstrap <- penguins %>%
  filter(species == "Chinstrap",
         !is.na(bill_length_mm),
         !is.na(bill_depth_mm))
chinstrap
## # A tibble: 68 × 8
##    species   island bill_length_mm bill_depth_mm flipper_length_mm body_mass_g
##    <fct>     <fct>           <dbl>         <dbl>             <int>       <int>
##  1 Chinstrap Dream            46.5          17.9               192        3500
##  2 Chinstrap Dream            50            19.5               196        3900
##  3 Chinstrap Dream            51.3          19.2               193        3650
##  4 Chinstrap Dream            45.4          18.7               188        3525
##  5 Chinstrap Dream            52.7          19.8               197        3725
##  6 Chinstrap Dream            45.2          17.8               198        3950
##  7 Chinstrap Dream            46.1          18.2               178        3250
##  8 Chinstrap Dream            51.3          18.2               197        3750
##  9 Chinstrap Dream            46            18.9               195        4150
## 10 Chinstrap Dream            51.3          19.9               198        3700
## # ℹ 58 more rows
## # ℹ 2 more variables: sex <fct>, year <int>

2B: Scatterplot

Make a scatterplot with the explanatory variable on the x-axis and the response variable on the y-axis. Add appropriate axis labels and a title.

ggplot(chinstrap, aes(x = bill_length_mm, y = bill_depth_mm)) +
  geom_point(alpha = 0.6) +
  labs(
    title = "Bill Length vs. Bill Depth (Chinstrap Penguins)",
    x = "Bill Length (mm)",
    y = "Bill Depth (mm)"
  )

Describe the association (direction, strength, linearity, any notable outliers):

There is a moderately strong, positive, roughly linear association between bill length and bill depth for Chinstrap penguins. Penguins with longer bills tend to also have deeper bills. The points follow a generally linear pattern with no major outliers.

2C: Correlation Coefficient

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

cor(chinstrap$bill_length_mm, chinstrap$bill_depth_mm)
## [1] 0.6535362

Interpretation of r:

The correlation coefficient is about 0.65, which indicates a moderately strong, positive, linear association between bill length and bill depth for Chinstrap penguins. Penguins with longer bills tend to also have deeper bills, and this pattern is fairly consistent across the sample.

2D: Fitting the Regression Line

Use lm() to fit the regression line. Extract the slope and intercept using coef() and write out the regression equation in the form y-hat = mx + b

model2 <- lm(bill_depth_mm ~ bill_length_mm, data = chinstrap)
coef(model2)
##    (Intercept) bill_length_mm 
##      7.5691401      0.2222117

Regression equation: 𝑦- hat = 0.22𝑥+7.57

Interpret the slope in context:

For each additional 1 mm of bill length, the model predicts bill depth to increase by about 0.22 mm, on average.

Interpret the intercept in context.:

The intercept represents the predicted bill depth when bill length is 0 mm. Because a bill length of 0 mm is not biologically possible for penguins, the intercept does not have a meaningful real‑world interpretation. It simply positions the regression line within the observed data range.

2E: Scatterplot with Regression Line

Add the regression line to your scatterplot.

ggplot(chinstrap, aes(x = bill_length_mm, y = bill_depth_mm)) +
  geom_point(alpha = 0.6) +
  geom_smooth(method = "lm", se = FALSE, color = "pink", linewidth = 1) +
  labs(
    title = "Bill Length vs. Bill Depth (Chinstrap Penguins)",
    x = "Bill Length (mm)",
    y = "Bill Depth (mm)"
  )
## `geom_smooth()` using formula = 'y ~ x'

2F: Making a Prediction and Calculating a Residual

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.

predict(model2, newdata = data.frame(bill_length_mm = 50))
##        1 
## 18.67973

The x value I chose: 50mm

Predicted y: 18.68

Find the actual y value for the first penguin in your filtered dataset. Compute its residual by hand (actual minus predicted). Interpret the sign of the residual — is this penguin above or below the regression line?

chinstrap$bill_depth_mm[1]
## [1] 17.9

Residual:

17.9−18.68=−0.78

Interpretation:

The residual is negative, meaning this penguin’s actual bill depth is below the regression line. In other words, its bill depth is slightly shallower than what the model predicts for a penguin with its bill length.