Introduction

Plane Crashes in San Francisco

Problem Statement

Hypothesis : Casualities is independent of flight phase.

• We are going to use cor-test and t-test for our hypothesis testing for this investigation.

• Cor test : A string indicating which correlation coefficient to calculate and test.cor. test() test for association/correlation between paired samples. It returns both the correlation coefficient and the significance level(or p-value) of the correlation 

• t-test : We use it to determine whether the means of two groups are equal to each other. The assumption for the test is that both groups are sampled from normal distributions with equal variances.

Data

• We collected data from the popular open data platform, Kaggle.
• The data was scrapped from the bureau of aircraft archives
• The population was enough in size did not need to be sampled. Hence no sampling method has been used.

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Data Cont.

• The important variables found through finding the correlation are, flight phase and region to the number of fatalities.
• Flight phase is a categorical variable ranging from 1-6.
• Flight’s period in air is of 5 phases.Taxi, Takeoff, Mid air,Descent and Parking
• The number of fatalities range from 0 to 550
• We cleaned the data, removed the nan and unknown values.

Descriptive Statistics and Visualisation

The important variables are total fatalities(0, 100, 200, 300, 400, 500) , flight phase(Flight landing, Parking, Takeoff, Taxing, Unknown).

From 1918 to 2019, more than 500 lives have been lost during flights and there were 300 fatalities due to landing problems like bad weather , collisions between planes or rough landings that caused injuries inside the aircraft. Around 70 deaths are caused by plane accidents in parking ,when the flight has landed and rest for parking ,due to some problems like fuel or if any plane strayed too close to the aerobridge and made contact will also lead to plane accidents in parking. Comparatively, Takeoff and landing have nearly identical percentages of deaths. . Basically taxiing means a plane to move along the ground under its own power, before takeoff. 335 people have died as a result of taxiing, either due to a lack of information about air taxiing or an engine failure.and finally 20 crashes for unknown cause. When we look at deaths by region, Asia leads with over 500 crashes and least in Oceania with 98 crashes.

We had a lot of missing data and null values , which we dealt with by first checking how many empty spaces were present, then replacing those empty spaces with na and further replacing na values with unknown for visualizing values better.

Plotting the number of fatalities according to the flight’s phase

#plotting the number of fatalities according to the flight's phase
ggplot(crash_analysis_df, aes(x = Flight_phase, y = Total_fatalities))+
  geom_bar(
    aes(fill = Total_fatalities), stat = "identity", color = "white",
    position = position_dodge(0.9)
  )

Plotting the number of fatalities according to region

#because we don't know which region the crash has occured we're categorising the factor world as unknown
crash_analysis_df[crash_analysis_df == "World"] <- 'Unknown'  
#plotting the number of fatalities according to region
ggplot(crash_analysis_df, aes(x = Region, y = Total_fatalities))+
  geom_bar(
    aes(fill = Total_fatalities), stat = "identity", color = "white",
    position = position_dodge(0.9)
  )

Decsriptive Statistics Cont.

• We used the knitr:kable function to print nice HTML tables.

#Calculating descriptive statistics for flight phase
crash_analysis_df1 %>% group_by(Flight_phase) %>% summarise(Min = min(Total_fatalities,na.rm = TRUE),
                                         Q1 = quantile(Total_fatalities,probs = .25,na.rm = TRUE),
                                         Median = median(Total_fatalities,na.rm = TRUE),
                                         Q3 = quantile(Total_fatalities,probs = .75,na.rm = TRUE),
                                         Max = max(Total_fatalities,na.rm = TRUE),
                                         Mean = mean(Total_fatalities,na.rm = TRUE),
                                         SD = sd(Total_fatalities,na.rm = TRUE),
                                         n = n(),
                                         Missing = sum(is.na(Total_fatalities))) -> table1
knitr::kable(table1)

Decsriptive Statistics Cont.

#Calculating descriptive statistics for region
crash_analysis_df1 %>% group_by(Region) %>% summarise(Min = min(Total_fatalities,na.rm = TRUE),
                                                            Q1 = quantile(Total_fatalities,probs = .25,na.rm = TRUE),
                                                            Median = median(Total_fatalities,na.rm = TRUE),
                                                            Q3 = quantile(Total_fatalities,probs = .75,na.rm = TRUE),
                                                            Max = max(Total_fatalities,na.rm = TRUE),
                                                            Mean = mean(Total_fatalities,na.rm = TRUE),
                                                            SD = sd(Total_fatalities,na.rm = TRUE),
                                                            n = n(),
                                                            Missing = sum(is.na(Total_fatalities))) -> table2
knitr::kable(table2)

Applying hypothesis

#applying cor.test on flight phase against the fatalities
cor.test(crash_analysis_df1$Flight_phase,crash_analysis_df1$Total_fatalities)

## 
##  Pearson's product-moment correlation
## 
## data:  crash_analysis_df1$Flight_phase and crash_analysis_df1$Total_fatalities
## t = -4.899, df = 28438, p-value = 9.684e-07
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.04064709 -0.01742238
## sample estimates:
##         cor 
## -0.02903866

Applying hypothesis Cont.

#applying t.test on flight phase against the fatalities
t.test(crash_analysis_df1$Flight_phase,crash_analysis_df1$Total_fatalities)

## 
##  Welch Two Sample t-test
## 
## data:  crash_analysis_df1$Flight_phase and crash_analysis_df1$Total_fatalities
## t = -34.545, df = 28778, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.634081 -3.243838
## sample estimates:
## mean of x mean of y 
##  2.142440  5.581399

Applying hypothesis Cont.

#applying cor.test on region against the fatalities
cor.test(crash_analysis_df1$Region,crash_analysis_df1$Total_fatalities)

## 
##  Pearson's product-moment correlation
## 
## data:  crash_analysis_df1$Region and crash_analysis_df1$Total_fatalities
## t = -10.124, df = 28438, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.07149628 -0.04833543
## sample estimates:
##         cor 
## -0.05992392

Applying hypothesis Cont.

#applying t.test on region against the fatalities
t.test(crash_analysis_df1$Region,crash_analysis_df1$Total_fatalities)

## 
##  Welch Two Sample t-test
## 
## data:  crash_analysis_df1$Region and crash_analysis_df1$Total_fatalities
## t = -5.7073, df = 29221, p-value = 1.159e-08
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.7662348 -0.3744825
## sample estimates:
## mean of x mean of y 
##  5.011041  5.581399

Hypothesis Testing

We applied Cor and t test for our hypothesis testing for this investigation.

We believe that the flight phase and region variables will show us as to which level has the most affect toward fatalities.

• For the variables Flight phase and Region we see that the Pearson’s product-moment gives the alernative hypothesis where the true correlation is not equal to 0 while the Welch Two Sample t-test states that the true difference in means is not equal to 0 , through this we can conclude that both the variables have a significant affect on the fatalities and that most fatalities occur during the flight and in the Asian regions.

Hypothesis Testing

model2 <- lm(Region ~ Flight_phase, data = crash_analysis_df1)
model2 %>% summary()

## 
## Call:
## lm(formula = Region ~ Flight_phase, data = crash_analysis_df1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.0969 -1.8714 -0.0218  0.9782  5.2790 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   5.172138   0.022518 229.688   <2e-16 ***
## Flight_phase -0.075193   0.009002  -8.353   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.96 on 28438 degrees of freedom
## Multiple R-squared:  0.002448,   Adjusted R-squared:  0.002413 
## F-statistic: 69.78 on 1 and 28438 DF,  p-value: < 2.2e-16

Hypthesis Testing Cont.

• Pearson correlation coefficient \[r = \frac{\Sigma(x_i-\bar{x})(y_i-\bar{y})}{\sqrt(\Sigma(x_i-\bar{x})^{2}\Sigma(y_i-\bar{y})^{2}) } \]

\[r = correlation\ coefficient\\ x_{i} = values\ of\ the\ x-variable\ in\ a\ sample\\ \bar{x} = mean\ of\ the\ values\ of\ the\ x-variable\\ y_{i} = values\ of\ the\ y-variable\ in\ a\ sample\\ \bar{y} = mean\ of\ the\ values\ of\ the\ y-variable\]

Hypthesis Testing Cont.

• Welch T-Test \[t = \frac{(m_A=m_B )}{\sqrt(S^{2}_A/n_A+S^{2}_B/n_B)}\] \[where,\ S_A\ and\ S_B\ are\ the\ standard\ deviation\ of\ the\ two\ groups\ A\ and\ B,\ respectively.\\Unlike\ the\ classic\ Student’s\ t-test,\ the\ Welch\ t-test\ formula\ involves\\ the\ variance\ of\ each\ of\ the\ two\ groups\ (S^{2}_A and S^{2}_B)\ being\ compared.\\ In\ other\ words,\ it\ does\ not\ use\ the\ pooled\ variances.\]

Discussion

• We started this investigation to check whether fatalities were independent of Flight phase and region which were proven wrong.
• We have strong hypothesis that according to region the fatalities differ, which could be an outcome of varying GDP’s of those accumulative regions or the service of airplanes hailing from that region.
• Going further through the investigation, it’d be great if we could co-relate how region and flight phase affect the fatalities in much deeper sense, so that a prediction could be made that makes sure people have safer journeys.
  
• Finally, we’d state that all the variables present in the new dataset affect the fatalities through one way or another but the flight’s phase(hinting towards whether the weather affects casualities) or the region(hinting toward whether the regions ability to provide optimal servicing) takes a toll on the casualities.

References

• ABE CEASAR PEREZ, Historical Plane Crash Data, Kaggle,last updated 26th May 2022, viewed 24th May 2022, https://www.kaggle.com/datasets/abeperez/historical-plane-crash-data.