Introduction

Finding the best location to stay while visiting Melbourne can be overwhelming; as each suburb around the city has a different range of options.
Airbnb listings can provide relevant information to compare and identify the best option.
This report will compare the difference between two independent sample means of Airbnb apartments in Fitzroy and St Kilda.

Problem Statement

Comparison of Airbnb entire apartment prices in Fitzroy and St. Kilda
Method:
- Two-sample t-test will be used to compare the independent Airbnb sample prices from Fitzroy and St. Kilda.
- Normality for each suburb will be evaluated, and scanned to identify any outliers.
- Homogeneity of Variance will be tested using Levene’s Test.

Data preprocessing

Dataset was obtained from Kaggle (sourced from “Inside Airbnb”), selecting Melbourne Airbnb listings; dated Jun-Jul 2018.
Before importing data to R, observations were preprocessed due to untidiness of the dataset.
Cluster sampling was applied by suburb (Fitzroy and St.Kilda), to select the sample observations.
Property type selected were only apartments.
Main variables kept were: suburb, state, property type, room type, price, review score.
No data values were modified on the preprocessing.

Data import to R

Main variables:
Suburb: Narrowed to Fitzroy and St. Kilda suburbs, this variable identifies the sampled price.
Price: Value of rent per day, ranging from $62 to $695 AUD.

setwd("C:/Users/karen/OneDrive/Documents/MC242/Intro to Statistics/Assignments/A3")

airbnb <- read_excel("airbnb.xlsx")

head(airbnb)

setwd( ) was used to set working directory so we could import and export the dataset to the specific directory right away
Import airbnb.xlsx data into R Studio
- Using readxl ( ) function from readxl package to import excel data set
- dataset will be saved under the name “airbnb” in R Studio
- head ( ) function enables us to see a small part of the observations

Data import to R cont..

Factors:
“Suburb” was converted to a factor, for which levels were “Fitzroy” and “St Kilda”.

# Set variable "suburb"" as factor ( )
airbnb$suburb <- airbnb$suburb %>% factor(levels=c("Fitzroy", "St Kilda"))

airbnb

The factor ( ) function was used to set “suburb” variable as factor
We can call out the airbnb dataset again to make sure suburb’s data type has been changed to “factor”

Descriptive Statistics

In this investigation, we are trying to focus on the daily price of Airbnb for Fitzroy area and St Kilda area
So, we are going to focus on the “suburb” and “price” variables

# Descriptive Statistics
desc_stats <- airbnb %>% group_by(suburb) %>% summarise(Min = min(price,na.rm = TRUE),
                                        Q1 = quantile(price,probs = .25,na.rm = TRUE),
                                        Median = median(price, na.rm = TRUE),
                                        Q3 = quantile(price,probs = .75,na.rm = TRUE),
                                        IQR = IQR(price, na.rm=TRUE),
                                        Max = max(price,na.rm = TRUE),
                                        Mean = mean(price, na.rm = TRUE),
                                        SD = sd(price, na.rm = TRUE),
                                        n = n(),
                                        Missing = sum(is.na(price))) 
kable(desc_stats)

suburb	Min	Q1	Median	Q3	IQR	Max	Mean	SD	n	Missing
Fitzroy	64	100	129	150	50	695	146.3655	85.82725	145	0
St Kilda	62	107	130	174	67	450	146.2552	60.43152	145	0

Descriptive Statistics:
- Using the group_by(suburb) we can group the descriptive statistics according to suburbs
- With the combination of summarise( ) function we calculate all the descriptive statistics (Min, Q1, Median, Q3, IQR, Max, Mean, SD, n and NAs) all together
The descriptive statistics will be saved under the name desc_stats, so we can call it out to show the result of the descriptive statistics for each suburb
The use of kable( ) function from knitr package is to show a nice table output

Visualization

# Box-plot:
boxplot(price ~ suburb, data = airbnb, ylab = "Suburb", xlab="Daily Price of an Entire Apartment", 
        main = "Entire Apartment Daily Price Comparison", horizontal = TRUE)

boxplot ( ) function is used to visualize the boxplot of Daily Price for Fitzroy and St Kilda, side by side in this case, so we can do a comparison.
Boxplot also includes outliers as we can see on the result above. In this case, we left the outliers as it is.

Hypothesis Testing

We selected Two-sample Independent T-test because we wanted to compare the daily prices of two independent suburbs
Hypotheses for two samples:

\[H_0: \mu_1 = \mu_2 (\mu_1 - \mu_2 = 0)\]

\[H_A: \mu_1 \ne \mu_2 (\mu_1 - \mu_2 \ne 0)\]

Assumption for Independent Two-Samples T-test are:
- Comparing two independent population means with unknown population variance.
- Whether population data are normally distributed or large sample (n > 30 for both groups)
- Homogeneity of Variance

Normality Assumption

airbnb_fitzroy <- airbnb %>% filter(suburb == "Fitzroy")
airbnb_fitzroy$price %>% qqPlot(dist="norm")

## [1] 69 82

airbnb_sk <- airbnb %>% filter(suburb == "St Kilda")
airbnb_sk$price %>% qqPlot(dist="norm")

## [1]  42 121

Using the filter( ) function, we can filter the suburb one by one
And with the combination of qqPlot( ) function, we can check the normality assumption for each suburb.
- The blue dashed lines corresponds to 95% CI for the normal quantiles.
- As we can see from the result, for both suburbs, there are some points that fall outside the blue dashed lines
- However, sample size of both groups are greater than 30, so we can assume normality for both cases

Homogeneity of Variance Assumption

\[H_0: \sigma_1^2 = \sigma_2^2 \]

\[H_A: \sigma_1^2 \ne \sigma_2^2 \]

leveneTest(price ~ suburb, data=airbnb)

leveneTest( ) function from car package was used to check the assumption of equal variance
- if p > 0.05, means fail to reject H0. In other words, we can assume equal variance and assumption is not violated
- if p < 0.05, means to reject H0. In other words, assumption is violated and the Homogeneity of Variance assumption does not hold. In this case an alternative test called the Welch Test can be used for two sample t-test with unequal variances
In this case, since p = 0.9052 is greater than 0.05 (significance level), we failed to reject H0 and the assumption of Homogeneity of Variance HOLDS.

Independent Two-Sample T-test

result <- t.test(price ~ suburb,
       data = airbnb,
       var.equal = TRUE,
       alternative = "two.sided"
       )
result

## 
##  Two Sample t-test
## 
## data:  price by suburb
## t = 0.012658, df = 288, p-value = 0.9899
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -17.04700  17.26769
## sample estimates:
##  mean in group Fitzroy mean in group St Kilda 
##               146.3655               146.2552

We can calculate the Independent Two-sample T-test using t.test( ) function.
- Since the Homogeneity of Variance assumption holds, we state var.equal = TRUE
- alternative = “two.sided” means it can either be lesser or greater
Independent Two-Sample T-test result summary:
- We assumed normality as both Fitzroy and St Kilda, n > 30
- LeveneTest p > 0.05 means variances for both groups are equal
- Estimated difference between means: x1 - x2 = 146.3655 - 146.2552 = 0.1103
- 95% CI [-17.04700 17.26769] captures H0 = 0.
- p-value = 0.9899, p > $\alpha$.
The decision is to fail to reject H_0

Discussion

Based on the Two Sample Independent T-test the decision is fail to reject the null hypothesis $H_0: \mu_1 = \mu_2 (\mu_1 - \mu_2 = 0)$ or Average Daily Price of Fitzroy’s Airbnb = Average Daily Price of St. Kilda’s Airbnb.
As p (0.9899) > 0.05, and the 95% CI [-17.04700 17.26769] captures the H0 = 0. Thus, the results of the test found no statistical significance on the comparison between mean prices of Airbnbs in Fitzroy and St. Kilda respectively.
As mentioned earlier, the Airbnb data was preprocessed and filtered into two popular Melbourne suburbs which is Fitzroy and St. Kilda. It was also narrowed down to renting enitre Apartments only instead of a Private Room or Shared Room.
However there were some limitations in the given dataset that could be improved for future investigations, a bigger sample size could give a more accurate mean price comparison between the two. In addition, the data was only for June and July 2018, a longer date range could possibly give a more accurate mean price of the Apartments depending on Seasonality/peak or off seasons.
Once again, based on the Two Sample Independent T-test there was no statistical significance between the Average daily price of Fitzroy and St.Kilda Airbnb.

Fitzroy or St.Kilda: Testing Airbnb’s price

MATH1324 Assignment 3

RPub link

Introduction

Problem Statement

Data preprocessing

Data import to R

Data import to R cont..

Descriptive Statistics

Visualization

Hypothesis Testing

Normality Assumption

Homogeneity of Variance Assumption

Independent Two-Sample T-test

Discussion

References