Hypotheses:

$H_0$: There is no relationship between the month and year of tornado occurrences (they are independent).
$H_1$: There is a relationship between the month and year of tornado occurrences (they are dependent).

Observed Frequencies (Contingency Table): $O = \begin{pmatrix} 26 & 41 & 87 & 97 \\ 2 & 41 & 46 & 63 \\ 13 & 25 & 18 & 225 \end{pmatrix}$

Row Sums: $R_1=251, R_2=152, R_3=281$. Column Sums: $C_1=41, C_2=107, C_3=151, C_4=385$. Grand Total: $N=684$.

Expected Frequencies: $E_{ij} = \frac{R_i \times C_j}{N}$

$E_{11} = \frac{251 \times 41}{684} \approx 15.06$
$E_{12} = \frac{251 \times 107}{684} \approx 39.30$
... (calculate all $E_{ij}$)

Chi-Square Test Statistic: $\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$

Degrees of Freedom: $df = (rows - 1)(cols - 1) = (3-1)(4-1) = 2 \times 3 = 6$

Critical Value: For $\alpha = 0.05$ and $df = 6$, $\chi^2_{critical} \approx 12.592$

R Code:


tornado_data <- matrix(c(26, 41, 87, 97,
                         2, 41, 46, 63,
                         13, 25, 18, 225),
                       nrow = 3, byrow = TRUE,
                       dimnames = list(Month = c("January", "February", "March"),
                                       Year = c("2015", "2014", "2013", "2012")))
chisq.test(tornado_data)
# Output will include:
# X-squared = 227.18, df = 6, p-value < 2.2e-16

Decision: Since $\chi^2_{calculated} (227.18) > \chi^2_{critical} (12.592)$ and p-value is very small, we reject $H_0$.

Conclusion: At $\alpha = 0.05$, there is sufficient evidence to conclude that a significant relationship exists between the month and year in which tornadoes occurred.

Question 3: Titanic Dataset Data Cleaning (R Script)

Dataset Download Link

https://raw.githubusercontent.com/datasciencedojo/datasets/master/titanic.csv

1. Familiarize with the dataset

Load and Initial Examination:


# Load necessary libraries
library(dplyr)
library(ggplot2)
library(VIM) # For missing data visualization

# Load the dataset
titanic_df <- read.csv("https://raw.githubusercontent.com/datasciencedojo/datasets/master/titanic.csv")

# Examine structure
str(titanic_df)
dim(titanic_df)
head(titanic_df)
tail(titanic_df)
summary(titanic_df)

Identify Data Types for Conversion:
- Survived: Needs to be a factor (0/1).
- Pclass: Needs to be a factor (1/2/3).
- Sex: Needs to be a factor.
- Embarked: Needs to be a factor.
- Age, Fare, SibSp, Parch: Should be numeric/integer.

Descriptive Statistics:


# Survival rates
mean(titanic_df$Survived) # Overall survival rate
titanic_df %>% group_by(Sex) %>% summarise(SurvivalRate = mean(Survived))
titanic_df %>% group_by(Pclass) %>% summarise(SurvivalRate = mean(Survived))

# Age and Fare distributions
summary(titanic_df$Age)
summary(titanic_df$Fare)

# Frequency tables for categorical variables
table(titanic_df$Sex)
table(titanic_df$Pclass)
table(titanic_df$Embarked)

# Check PassengerId for uniqueness
length(unique(titanic_df$PassengerId)) == nrow(titanic_df)

# Examine sample Name and Ticket entries
head(titanic_df$Name)
head(titanic_df$Ticket)

# Initial visualizations
hist(titanic_df$Age, main = "Distribution of Age", xlab = "Age")
hist(titanic_df$Fare, main = "Distribution of Fare", xlab = "Fare")

2