Assignment
Justin Sia
2024-04-05
1. Present the study design, null and alternative hypotheses (5%)
Study Design - A cross-sectional investigation of house conditions in 372 suburbs was conducted to determine whether the housing prices varied among houses with different numbers of rooms.
Null Hypothesis - Housing prices does not vary among houses with differnt numbers of rooms.
Alternative Hypothesis - Housing prices does vary among houses with different numbers of rooms
2.Describe the characteristics of the study sample by the number of rooms/house (10%)
2.1 Load the Dataset
f = file.choose()
housing_prices = read.csv(f)2.2 Charcteristics of the study
library(table1)## Warning: package 'table1' was built under R version 4.2.3##
## Attaching package: 'table1'## The following objects are masked from 'package:base':
##
## units, units<-table1(~ river + price + age + industry + ptratio + low_ses | rooms, data = housing_prices)| 4-room (N=15) |
5-room (N=27) |
6-room (N=254) |
7-room (N=76) |
Overall (N=372) |
|
|---|---|---|---|---|---|
| river | |||||
| No | 15 (100%) | 23 (85.2%) | 239 (94.1%) | 67 (88.2%) | 344 (92.5%) |
| Yes | 0 (0%) | 4 (14.8%) | 15 (5.9%) | 9 (11.8%) | 28 (7.5%) |
| price | |||||
| Mean (SD) | 17.3 (10.7) | 14.0 (5.05) | 18.9 (5.50) | 28.8 (11.7) | 20.5 (8.59) |
| Median [Min, Max] | 13.8 [7.00, 50.0] | 14.4 [5.00, 23.7] | 19.4 [5.00, 50.0] | 27.5 [7.50, 50.0] | 19.8 [5.00, 50.0] |
| age | |||||
| Mean (SD) | 93.5 (15.9) | 89.8 (20.4) | 75.6 (24.3) | 77.2 (21.1) | 77.7 (23.6) |
| Median [Min, Max] | 100 [37.8, 100] | 96.2 [9.80, 100] | 84.5 [6.00, 100] | 82.7 [2.90, 100] | 87.3 [2.90, 100] |
| industry | |||||
| Mean (SD) | 17.8 (2.23) | 17.7 (5.43) | 13.6 (6.16) | 11.1 (6.53) | 13.5 (6.32) |
| Median [Min, Max] | 18.1 [9.90, 19.6] | 18.1 [6.91, 27.7] | 13.9 [2.18, 27.7] | 9.90 [1.89, 19.6] | 18.1 [1.89, 27.7] |
| ptratio | |||||
| Mean (SD) | 19.3 (1.94) | 18.7 (2.31) | 19.3 (1.71) | 18.4 (1.81) | 19.1 (1.82) |
| Median [Min, Max] | 20.2 [14.7, 20.2] | 20.1 [14.7, 21.2] | 20.2 [14.7, 21.2] | 18.0 [14.7, 21.0] | 20.2 [14.7, 21.2] |
| low_ses | |||||
| Mean (SD) | 24.4 (11.4) | 23.3 (6.75) | 14.5 (5.37) | 9.14 (6.25) | 14.4 (7.15) |
| Median [Min, Max] | 29.3 [3.26, 38.0] | 24.0 [10.2, 34.4] | 14.1 [5.08, 34.0] | 6.79 [1.73, 25.8] | 13.6 [1.73, 38.0] |
The data is predominantly 6-room houses, covering 68% of the all samples. Generally speaking, the mean house price increases as number of rooms increases, from an average of $17,300 for 4-room to $28,800 for 7-room houses. There is wide price range in each room category, with the largest range being the 6-room houses, ranging from $5,000 to $50,000.
3 Develop and interpret a histogram for the distribution of house prices (10%)
library(ggplot2)## Warning: package 'ggplot2' was built under R version 4.2.3p = ggplot(data = housing_prices, aes(x = price))
p1 = p + geom_histogram(aes(y = ..density..), color = "white", fill = "blue")
p2 = p1 + geom_density(col="red")
p2 + ggtitle("Distribution of housing prices") + theme_bw()## Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
## ℹ Please use `after_stat(density)` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
The histogram shows that the housing prices is positively skewed, with quite a long tail to the right, indicating there are a decent number of houses with very high housing prices. Though the bulk of the data lies relatively normally distributed around the peak of the $20,000 range, with a roughly symmetric, bell-shaped curve.
4 Develop and interpret a box plot to describe the differences in housing prices among the number of rooms per household (10%)
p = ggplot(data = housing_prices, aes(x = rooms, y = price, fill = rooms, col = rooms))
p1 = p + geom_boxplot(col = "black") + geom_jitter(alpha = 0.05)
p1 + labs(x = "Rooms", y = "Housing Prices (USD)") + ggtitle("Housing Prices by Rooms") + theme_bw()
Once again, there is a overall trend of increaing median price as the number of rooms increases. The variation in prices, as indicated by the size of the boxes and whiskers, also tends to increase with the number of rooms.For all room categories, the median is closer to the lower end of the box, indicating right-skewness in the price distributions. This skewness appears most pronounced for 7-room houses. The outliers seem to mostly of the highly priced houses from the tail observed in the distribution graph seems to mostly come from the 6-7 room houses.
5. Conduct a statistical test to determine whether housing prices were different among houses with different numbers of rooms. Interpret the findings (15%)
kruskal.test(price ~ rooms, data = housing_prices)##
## Kruskal-Wallis rank sum test
##
## data: price by rooms
## Kruskal-Wallis chi-squared = 75.504, df = 3, p-value = 2.826e-16The very low p-value (2.826e-16) suggests that we have extremely strong evidence to reject the null hypothesis, which means there is a statistically significant difference in housing prices among the different room categories.The Kruskal-Wallis test is rank-based, which means it’s less sensitive to outliers and doesn’t assume normality of the data within each group, making it a robust choice for this analysis. Since Kruskal-Wallis is an omnibus test, we can only conclude that at least one room category having a significantly different median price than the others. This supports the findings from the descriptive analysis and visualization, which showed increasing median prices with an increasing number of rooms.
6. Determine which particular number of rooms/house had different house prices using the Tukey posthoc test. Fill in the following table and interpret the findings (20%)
price.rooms = aov(price ~ rooms, data = housing_prices)
summary(price.rooms)## Df Sum Sq Mean Sq F value Pr(>F)
## rooms 3 7162 2387.2 43.49 <2e-16 ***
## Residuals 368 20202 54.9
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1par(mfrow=c(2,2))
plot(price.rooms)
tukey.price.rooms = TukeyHSD(price.rooms)
tukey.price.rooms## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = price ~ rooms, data = housing_prices)
##
## $rooms
## diff lwr upr p adj
## 5-room-4-room -3.215556 -9.373321 2.942210 0.5331459
## 6-room-4-room 1.603780 -3.477109 6.684668 0.8475492
## 7-room-4-room 11.511053 6.108559 16.913546 0.0000004
## 6-room-5-room 4.819335 0.948716 8.689954 0.0077590
## 7-room-5-room 14.726608 10.442544 19.010672 0.0000000
## 7-room-6-room 9.907273 7.407162 12.407384 0.0000000