• Airlines and passengers both care about on-time flights.
• Question: Do longer flights tend to have longer arrival delays?
• hllo
• We will use a small simulated data-set of flights with:
  • distance (in miles)
  • seats (number of seats on the plane)
  • arr_delay (arrival delay in minutes)
• Goal: use linear regression to model the relationship between distance and arrival delay.

We model arrival delay as a linear function of distance:
\[arr\_delay = \beta_0 + \beta_1 * distance + \epsilon\]

where:

• \(\beta_0: intercept\ (baseline\ delay\ when\ distance = 0)\)

• \(\beta_1: change\ in\ delay\ (minutes)\ for\ each\ extra\ mile\ of\ distance\)

• \(\epsilon: random\ error\ (other\ factors:weather,\ traffic,\ etc.)\)

We will estimate \(\beta_0\) and \(\beta_1\) from our flight data.

Here we create a small dataset of 50 flights

n <- 50

distance <- runif(n, min = 200, max = 2500) # miles
seats <- sample(c(100, 150, 180, 220), size = n, replace = TRUE)

#true relationship: base 5 min delay + 0.01 min per mile + noise

arr_delay <- 5 + 0.01 * distance + rnorm(n, mean = 0, sd = 10)

flights <- data.frame(distance, seats, arr_delay)

• arr_delay is the response (Y)

• distance and seats are aviation related predictors (X)

• Each point = one flight

• The smooth line is fitted regression line

We fit the model:

\[arr\_delay = \beta_0 + \beta_1 * distance + \epsilon\]

Important output:

• Estimated slope \(\hat{\beta}_1 =\) 0.0088: each additional mile adds ~0.009 minutes of delay

• p-value for \(\hat{\beta}_1 =\) 0.000033: strong evidence of a relationship between distance and delay

• \(R^2 =\) 0.304: about 30% of the variability in arrival delay is explained by distance

Suppose the fitted model is:

\[\hat{arr\_delay} = \hat{\beta}_0 + \hat{\beta}_1 * distance\]

Interpretation of \(\hat{\beta}_1\):

• Estimated change in arrival delay (minutes) for a one-mile increase in distance

Example:

• With \(\hat{\beta}_1 = 0.0088\), a 100-mile increase in distance adds about

  \[100 * 0.0088 = 0.88\ minutes\]

  to the expected arrival delay (on average)

Residuals check whether the linear model is reasonable:

\[e_i = y_i - \hat{y}_i\]

• Residuals should be randomly scattered around 0.

• A strong pattern (curve or trend) would suggest the linear model may not be adequate.

• Points far from 0 may indicate unusual flights (potential outliers).

• If the vertical spread grows with fitted values, it could mean non-constant variance.

Now add a second aviation variable: plane size (seats).

• Rotate the plot to see how delay changes with both distance and seats

• We used simple linear regression to study flight delays.

• Longer flights tended to have different expected arrival delays than shorter flights.

• Linear regression helps us:

  • Quantify relationships between aviation variables

  • make predictions for new flights

  • Check model assumptions using residual plots.