Import dataset
housing_prices = read.csv("C:/Users/24864833/OneDrive - UTS/Desktop/Housing_prices.csv")Number of variables and their names
dim(housing_prices)## [1] 372 8names(housing_prices)## [1] "ID" "river" "rooms" "price" "age" "industry" "ptratio"
## [8] "low_ses"Describe the characteristics of the study sample by the number of rooms/house
library(table1)##
## Attaching package: 'table1'## The following objects are masked from 'package:base':
##
## units, units<-table1(~ price + river + 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) |
|
|---|---|---|---|---|---|
| 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] |
| 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%) |
| 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] |
Develop and interpret a histogram for the distribution of house prices (10%)
library(ggplot2)
library(grid)
library(gridExtra)
library(dplyr)##
## Attaching package: 'dplyr'## The following object is masked from 'package:gridExtra':
##
## combine## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(GGally)## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2library(ggthemes)
p = ggplot(data = housing_prices, aes(x = price)) + geom_histogram(color = "black", fill = "light blue") + labs(x = "Median housing prices (x USD 1,000)") + labs(y = "Count number")
p1 = ggplot(data = housing_prices, aes(x = price, y = ..density..)) + geom_histogram(color = "black", fill = "light blue") + geom_density(col="blue") + labs(x = "Median housing prices (x USD 1,000)") + labs(y = "Density")
grid.arrange(p, p1, nrow = 2, top = textGrob("Distribution of house prices", gp = gpar(fontsize = 18, font = 2)))## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## 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`.
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 = "cyan") + geom_jitter(alpha = 0.01) + labs(x = "Room numbers of each household", y = "Median housing prices (x USD 1,000)") + ggtitle("Box Plot of housing prices by room numbers of each household") + theme_bw()
p1
Conduct a statistical test to determine whether housing prices were
different among houses with different numbers of rooms. Interpret the
findings (15%) Kruskal-Wallis test
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-16ANOVA test
anova_test= aov(price ~ rooms, data = housing_prices)
summary(anova_test)## 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 ' ' 1The Kruskal-Wallis and ANOVA tests yielded extremely low p-values (2.826e-16 and 2e-16), both less than 0.05. Thus, we reject the null hypothesis (H0) in favor of the alternative hypothesis (Ha).
In summary, there are statistically significant differences in housing prices across houses with varying numbers of rooms
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%)
tukey_result = TukeyHSD(anova_test)
tukey_result## 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 0.0000000tukey_results <- data.frame(
Comparisons = c("5-room houses vs. 4-room houses","6-room houses vs. 4-room houses", "7-room houses vs. 4-room houses", "6-room houses vs. 5-room houses", "7-room houses vs. 5-room houses", "7-room houses vs. 6-room houses"),
Mean_Differences_in_housing_price= c("-3.21 (-9.37, 2.94)","1.60 [-3.48, 6.68]","11.51 [6.11, 16.91]","4.82 [0.95, 8.69]","14.73 [10.44, 19.01]","9.91 [7.41, 12.41]"),
P_Value = c("0.5331", "0.8475", "0.0000", "0.0078", "0.0000", "0.0000")
)
print(tukey_results, row.names = FALSE)## Comparisons Mean_Differences_in_housing_price P_Value
## 5-room houses vs. 4-room houses -3.21 (-9.37, 2.94) 0.5331
## 6-room houses vs. 4-room houses 1.60 [-3.48, 6.68] 0.8475
## 7-room houses vs. 4-room houses 11.51 [6.11, 16.91] 0.0000
## 6-room houses vs. 5-room houses 4.82 [0.95, 8.69] 0.0078
## 7-room houses vs. 5-room houses 14.73 [10.44, 19.01] 0.0000
## 7-room houses vs. 6-room houses 9.91 [7.41, 12.41] 0.0000The Tukey posthoc analysis indicates that houses with 7 rooms exhibit significant differences in housing prices (P-value ≈ 0 <0.05) compared to those with 4, 5, and 6 rooms. Similarly, significant differences are observed when comparing houses with 6 rooms to those with 5 rooms (P-value = 0.0078 <0.05).
Conversely, when comparing houses with 5 and 6 rooms to those with 5 rooms alone, there are no statistically significant differences in house prices (P-value=0.53 and 0.85 > 0.05).