There is a total of 12 variables. I consider 5 meaningful variables suitable for simple linear regression and hypothesis testing.

• Survived
  • 0 = Didn’t survive
  • 1 = Survived
• Pclass
  • Passenger’s ticket class, represent a passenger’s economic status
  • 1 = First class
  • 2 = Second class
  • 3 = Third class
• Sex
  • “male” and “female”
• Age
  • Passengers’ ages
• Fare
  • Ticket price that passengers paid

Simple Linear Regression is the relationship between 2 quantitative variables. It is expressed by this equation:
\[ y = \beta_0 + \beta_1 x + \epsilon \] For the first analysis, I’ll be first exploring the correlation between Survived and Fare. The model of regression in this case will be: \[ Survived = \beta_0 + \beta_1 \times Fare \] After calculating the values, which will be displayed next page, we get: \[ Survived = 0.3197 + 0.0025 \times Fare \]

Below is the R code and its output for the linear regression model:

fare_LRM <- lm(Survived ~ Fare, data = ttdf_filtered)
#summary(fare_LRM)

Intercept = 0.30270, means when fare is 0, the survival probability is arouond 30.27%. Fare coefficient = 0.00252, which means 1 unit increase in fare will result in the increase of surviving by 0.252%

confint(fare_LRM)

##                   2.5 %      97.5 %
## (Intercept) 0.278114862 0.361384438
## Fare        0.001832389 0.003148964

#summary(fare_LRM)
t_alpha <- qt(0.975, df = 712)
t_alpha

## [1] 1.963301

\[ \text{CI} = \beta_1 \pm t_{\alpha/2} \times \text{SE}(\beta_1) \] From the result we get by the code summary, we can get

\[ \text{CI} = 0.0024907 \pm \text{1.963301} \times(0.0003353) \] \[ = [0.001832,0.003149] \]

Confidence Intervals: Under 95% confidence level, the true value of intercept is about 0.2781 ~ 0.3614; the true value of Fare’s coefficient is about 0.0018 ~ 0.0031. Both are positive significant.

P-value: For both the intercept (< 2e-16) and Fare (3.16e-13), the P-values are extremely small, which means both are statistically significant. Moreover, the P-value of Fare is way less than 0.05, so we have enough confidence to prove that Fare has a significant effect on survival probability.

## `geom_smooth()` using formula = 'y ~ x'

## `geom_smooth()` using formula = 'y ~ x'

As we can easily find out, people who are female or living in higher passenger class were easier to survive.

The reason that I pick “Fare” and “Age” is because the values of these two have more differences between each other, compare to Pclass, which is only in the scale of 1~3.