Data Dive: Confidence Intervals

Introduction

This data dive examines the relationship between original numeric variables and newly constructed variables derived from them. The goal is to assess how variable definitions, transformations, and documentation influence interpretation, model building, and inference.

Pair 1: Arrival Delay vs. Delay Recovery

library(nycflights13)
library(tidyverse)

set.seed(192321)

df <- flights |>
  filter(!is.na(dep_delay), !is.na(arr_delay)) |>
  mutate(delay_recovery = dep_delay - arr_delay)
df |>
  ggplot(aes(x = delay_recovery, y = arr_delay))+
  geom_point(alpha = 0.2)+
  labs(
    title = "Arrival Delay vs Delay Recovery",
    x = "Delay Recovery (dep_delay - arr_delay)",
    y = "Arrival Delay (minutes)"
  ) +
  theme_classic()

Interpretation

The plot shows a strong negative linear relationship between arrival delay and delay recovery. However, this relationship is mechanically induced because delay recovery is defined as dep_delay - arr_delay. Since arrival delay is directly embedded in the construction of delay recovery, the linear pattern observed in the visualization reflects algebraic structure rather than an independent empirical relationship. This highlights the importance of documenting derived variables when interpreting model results.

Scrutiny of the Plot

Several extreme arrival delay values are visible in the upper portion of the plot, representing flights with very large delays. These points could strongly influence statistical measures such as correlation and should be examined carefully before fitting any formal model. Additionally, the triangular shape of the plot reflects algebraic constraints between the variables rather than purely empirical structure. This reinforces the importance of distinguishing structural relationships from true anomalies in the data.

Correlation

cor(df$arr_delay, df$delay_recovery)
## [1] -0.4423213

Interpretation

The correlation between arrival delay and delay recovery is approximately −0.44, indicating a moderate negative linear relationship. This negative association is expected because delay recovery is defined as the difference between departure and arrival delay. However, the relationship is not perfectly linear due to variation in departure delays and the presence of extreme delay values. This demonstrates that while the variables are mathematically related, real-world variability weakens the strength of the linear association.

Confidence Interval for Arrival Delay

t.test(df$arr_delay)$conf.int
## [1] 6.742478 7.048276
## attr(,"conf.level")
## [1] 0.95

Interpretation

The 95% confidence interval for the mean arrival delay is approximately (6.74, 7.05) minutes. Because this interval is narrow, we can estimate the population mean arrival delay with high precision. This reflects the large sample size of the dataset. Although extreme delays occur, the average arrival delay across all flights is slightly under 7 minutes.

Pair 2: Air Time vs. Speed

df2 <- flights |>
  filter(!is.na(distance), !is.na(air_time)) |>
  mutate(speed = distance / air_time * 60)
df2 |>
  ggplot(aes(x = air_time, y = speed)) +
  geom_point(alpha = 0.2) +
  labs(
    title = "Air Time vs. Calculated Speed",
    x = "Air Time (minutes)",
    y = "Average Speed (mph)"
  ) +
  theme_classic()

Interpretation

The relationship between air time and calculated speed exhibits clear nonlinear structure. Because speed is defined as distance divided by air time, air time appears in the denominator of the constructed variable, introducing a mechanical dependency. The curved bands visible in the plot correspond to different route distances, each producing a hyperbolic relationship between air time and speed.

Shorter flights show greater variability in calculated speed because takeoff and landing time represent a larger proportion of total air time. In contrast, longer flights tend to cluster around more stable cruising speeds.

Scrutiny of the Plot

Several unusually high speed values are visible in the upper portion of the plot. These may represent data entry issues, extremely short air times, or measurement noise. Before fitting a formal model, these extreme observations should be investigated to determine whether they reflect true values or potential errors.

Correlation

cor(df2$air_time, df2$speed)
## [1] 0.6144748

Interpretation

The correlation between air time and calculated speed is approximately 0.61, indicating a moderate-to-strong positive linear relationship. This suggests that flights with longer air times tend to have higher average speeds. This pattern is reasonable because shorter flights include proportionally more time spent taking off and landing, which lowers average speed, while longer flights spend more time at cruising altitude where speeds are higher and more stable.

Confidence Interval for Speed

t.test(df2$speed)$conf.int
## [1] 394.0659 394.4814
## attr(,"conf.level")
## [1] 0.95

Interpretation

The 95% confidence interval for the mean flight speed is approximately (394.07, 394.48) mph. Because this interval is extremely narrow, we can estimate the population mean speed with high precision. This reflects the large sample size and the relatively consistent cruising speeds across flights. Despite variation among individual routes, the average flight speed across NYC departures is approximately 394 mph.

Conclusion

This data dive demonstrates the importance of documenting how variables are constructed before interpreting statistical relationships. In Pair 1, the observed negative association between arrival delay and delay recovery was mechanically induced because the explanatory variable was directly derived from the response variable. This highlights how strong correlations can arise from algebraic structure rather than independent empirical relationships.

In Pair 2, the relationship between air time and calculated speed reflected both mathematical construction and meaningful real-world aviation patterns. The moderate positive correlation and narrow confidence interval for mean speed suggest stable cruising behavior across flights, while also illustrating how nonlinear structure may not be fully captured by linear correlation.

Overall, this investigation reinforces that statistical measures such as correlation and confidence intervals must be interpreted in the context of variable definitions and data documentation. Without careful documentation, mechanical relationships may be mistaken for substantive findings, leading to misleading conclusions in modeling and inference.