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:

I decided to choose option 2

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(body_mass_g))

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 = body_mass_g)) +
     geom_point(alpha = 0.6) +
     labs(
         title = "Bill Length vs. Body Mass (Chinstrap Penguins)",
         x     = "Bill Length (mm)",
         y     = "Body Mass (g)"
     )

 cor(Chinstrap$bill_length_mm, Chinstrap$body_mass_g, use = "complete.obs")
## [1] 0.5136383
 model <- lm(body_mass_g ~ bill_length_mm, data = Chinstrap)

  coef(model)
##    (Intercept) bill_length_mm 
##      846.14333       59.11773

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

There is a moderately weak association between bill length and body mass for Chinstrap penguins. Penguins with longer bills seem to have a greater body mass. There are a few outliers in our scatter plot graph.

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$bill_length_mm, Chinstrap$body_mass_g, use = "complete.obs")
## [1] 0.5136383

Interpretation of r:

We see the correlation coefficient of 0.51 can support the positivity of our scatter plot. and the magnitude shows that there is a moderate strength (slightly more on the stronger side) between the variables. penguins with longer bills seem to be heavier, and this is somewhat consistant throughout the graph.

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

model <- lm(body_mass_g ~ bill_length_mm, data = Chinstrap)
coef(model)
##    (Intercept) bill_length_mm 
##      846.14333       59.11773

Regression equation:

y-hat = 59.1x + 846, x is bill lentgh in mm and y-hat is predicted body mass in grams.

Interpret the slope in context:

for every millimeter of bill length, the penguin will be around 59.1 grams heavier.

Interpret the intercept in context.:

The intercept claims that if a penguin had a bill length of 0 mm, than its body mass would be 846 grams. This is kind of weird because if a penguin had a bill length of 0 it probably wouldnt live

2E: Scatterplot with Regression Line

Add the regression line to your scatterplot.

ggplot(Chinstrap, aes(x = bill_length_mm, y = body_mass_g)) +
  geom_point(alpha = 0.6) +
  geom_smooth(method = "lm", se = FALSE, color = "darkgreen", linewidth = 1) +
  labs(
    title    = "Bill Length vs. Body Mass (Chinstrap Penguins)",
    x        = "Bill Length (mm)",
    y        = "Body Mass (g)"
  )
## `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.

59.11773*50.1 - 846.14333
## [1] 2115.655

The x value I chose: 50.1, It shared the bill length of one of the chinstrap penguins

Predicted y: 2115.655

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?

3400-2115.655
## [1] 1284.345

Residual:

1284.345

Interpretation:

The residual is about 1284 grams. This is a very high positive number showing that this penguin is 1284 grams more than what the y-hat predicts for a penguin of its bill length. However it is still below the regression line