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 chose option 1.

2A: Filter and Describe Your Data

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

gentoo<-penguins %>%
  filter(species=="Gentoo", 
         !is.na(bill_depth_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(gentoo, aes(x=bill_depth_mm, y=body_mass_g)) +
  geom_point(alpha = 0.6) +
  labs(
    title = "Bill Depth vs Body Mass of Gentoo Penguins", 
    x= "Bill Depth (mm)",
    y= "Body Mass (g)"
  )

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

The graph in positive and a moderately strong association. It is linear. There is one noticeable outier at (15.25mm, ~6300g)

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

Interpretation of r:

r is above 0, so its a positive association, and close to 1, so the the association is also fairly strong.

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_depth_mm, data = gentoo)
coef(model)
##   (Intercept) bill_depth_mm 
##     -458.9852      369.4406

Regression equation:

y-hat = 369.4406 x + -458.9852

Interpret the slope in context:

For every 1mm the bill depth of a penguin increases, the body mass will increase by 369.4406g on average.

Interpret the intercept in context.:

At 0mm of bill depth, the body mass will be -458.9852g. This is not applicable to real world scenerios.

2E: Scatterplot with Regression Line

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="purple", linewidth = 1) +
  labs(
    title = "Bill Depth vs Body Mass of Gentoo Penguins",
    x= "Bill Depth (mm)",
    y= "Body Mass(g)"
  )
## `geom_smooth()` using formula = 'y ~ x'

2F: Making a Prediction and Calculating a Residual

Choose a point of (x,y) that sits in your scatterplot. Use your linear model to predict the y-hat value that corresponds with this x-value. Report and interpret the result.

369.4406*14-458.9852
## [1] 4713.183

The x value I chose: 15 from (15,4750)

Predicted y: 4713.183

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?

4750-4713.183
## [1] 36.817

Residual:

36.817

Interpretation:

The residual is a positive value, so the predicted value was smaller than the actual value. So at 15mm for bill depth, the model predicts a value lower by 36.817g for the body weight as compared to an actual penguin with a bill depth of 15mm.