Introduction

For this project, I am working with the faithful dataset, which is available in R. The data comes from the Old Faithful geyser in Yellowstone National Park and records how the geyser behaves over time. Each observation includes two pieces of information:

How long an eruption lasted (in minutes)

How long people had to wait until the next eruption (in minutes)

My goal is to explore whether these two variables are connected. In other words, does the length of an eruption tell us anything about how long the waiting time will be afterward?

To answer this question, I will look at the basic structure of the data, create visualizations, and build simple regression models. Through this analysis, I hope to understand the pattern between the eruption duration and the waiting time in a clear and meaningful way.

Hypothesis: I think that longer eruptions lead to longer waiting times. It seems logical that if the geyser erupts for a long time, it might need more time to build up pressure before the next eruption.

Methods

Below are the steps I followed:

  1. Load and inspect the dataset
  2. Visualize the relationship between the two variables
  3. Fit regression models to test the hypothesis

Results

##Loading and inspecting the data##


data("faithful")
head(faithful)
##   eruptions waiting
## 1     3.600      79
## 2     1.800      54
## 3     3.333      74
## 4     2.283      62
## 5     4.533      85
## 6     2.883      55
summary(faithful)
##    eruptions        waiting    
##  Min.   :1.600   Min.   :43.0  
##  1st Qu.:2.163   1st Qu.:58.0  
##  Median :4.000   Median :76.0  
##  Mean   :3.488   Mean   :70.9  
##  3rd Qu.:4.454   3rd Qu.:82.0  
##  Max.   :5.100   Max.   :96.0

The dataset has 272 observations. Eruptions last about 1.6 to 5.1 minutes, and waiting times range from 43 to 96 minutes.

Visualisation

This plot shows how the two variables relate. The regression line helps show the trend.

install.packages("ggplot2")
library(ggplot2) # for data visualization

ggplot(faithful,
       aes
       (x = eruptions,
         y = waiting)) +
geom_point(alpha = 0.6) +
geom_smooth(method = "lm", se = TRUE) +
labs(
x = "Eruption Duration (minutes)",
y = "Waiting Time (minutes)",
title = "Relationship Between Eruption Duration and Waiting Time"
)
## `geom_smooth()` using formula = 'y ~ x'

There is clearly a positive pattern.In simple words, when the eruption lasts longer, the time before the next eruption also gets longer. The dots are spread a bit around the line, which means the connection is clear but not perfect, there are some differences in the data.

Simple Linear Regression

This model tests whether eruption duration predicts waiting time.

lm1 = lm(waiting ~ eruptions, data = faithful)
summary(lm1)
## 
## Call:
## lm(formula = waiting ~ eruptions, data = faithful)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -12.0796  -4.4831   0.2122   3.9246  15.9719 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  33.4744     1.1549   28.98   <2e-16 ***
## eruptions    10.7296     0.3148   34.09   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.914 on 270 degrees of freedom
## Multiple R-squared:  0.8115, Adjusted R-squared:  0.8108 
## F-statistic:  1162 on 1 and 270 DF,  p-value: < 2.2e-16
confint(lm1)
##                2.5 %   97.5 %
## (Intercept) 31.20069 35.74810
## eruptions   10.10996 11.34932

The slope, shown by the eruption coefficient of 10.73, means that for every additional minute of eruption, the waiting time increases by about 10.7 minutes. The p-value (< 2e-16) shows that this relationship is statistically significant.

The confidence intervals show that the intercept is between 31.20 and 35.75, and the slope is between 10.11 and 11.35. This means the waiting time increases by about 10 to 11 minutes for each extra minute of eruption, confirming a strong and significant positive relationship.

Residuals for the Simple Linear Regression

This code produces residual diagnostics for the first model (linear). It helps us check normality, equal variance, and outliers.

install.packages("ggfortify")
library(ggfortify) # to use autopilot() function for diagnostic plots


autoplot(lm1)

The plot shows that the data doesn’t follow a straight-line pattern and the spread of points changes across the graph. This means the model doesn’t fit the data perfectly. To fix this, we can add a polynomial term, which helps the line bend and match the data better.

#Polynomial Regression

lm2 = lm(waiting ~ poly(eruptions, 2), data = faithful)
summary(lm2)
## 
## Call:
## lm(formula = waiting ~ poly(eruptions, 2), data = faithful)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -12.6471  -4.0904   0.2855   3.5204  14.6787 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          70.8971     0.3503  202.41  < 2e-16 ***
## poly(eruptions, 2)1 201.6029     5.7767   34.90  < 2e-16 ***
## poly(eruptions, 2)2 -21.6025     5.7767   -3.74 0.000225 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.777 on 269 degrees of freedom
## Multiple R-squared:  0.8208, Adjusted R-squared:  0.8194 
## F-statistic:   616 on 2 and 269 DF,  p-value: < 2.2e-16
autoplot(lm2)

After adding the polynomial term, the model improved. The residuals are now more evenly spread, showing better linearity and less heteroscedasticity. The Q-Q plot looks closer to normal, and there are no major outliers, meaning the model fits the data much better.

Conclusion and Reflection

The analysis of the faithful dataset shows a strong positive link between eruption duration and waiting time. When the eruption lasts longer, the waiting time before the next one also increases. The simple linear model proved this relationship, but the residual plots showed that the data wasn’t perfectly linear. After adding a polynomial term, the model became a better fit, with residuals more evenly spread and closer to normal. Overall, the polynomial model gave a clearer and more accurate picture of the Old Faithful geyser’s behavior.

By doing this project the most challenging part was finding the proper data. I have worked with one data for very log time, but it was very hard and challenging to continue it without instructor assistance. This project even though is not very challenging requires much time and attention to details. One important thing that I have learnt while doing this project is that the R is so comprehensive and offers solution to any issue that might arise with the data but for that one need to have strong expertise in it.