Question-a
To begin, load in the Boston data set.The Boston data set is part of the MASS library in R. How many rows are in this data set? How many columns? What do the rows and columns represent?:
library(MASS)
head(Boston)## crim zn indus chas nox rm age dis rad tax ptratio black
## 1 0.00632 18 2.31 0 0.538 6.575 65.2 4.0900 1 296 15.3 396.90
## 2 0.02731 0 7.07 0 0.469 6.421 78.9 4.9671 2 242 17.8 396.90
## 3 0.02729 0 7.07 0 0.469 7.185 61.1 4.9671 2 242 17.8 392.83
## 4 0.03237 0 2.18 0 0.458 6.998 45.8 6.0622 3 222 18.7 394.63
## 5 0.06905 0 2.18 0 0.458 7.147 54.2 6.0622 3 222 18.7 396.90
## 6 0.02985 0 2.18 0 0.458 6.430 58.7 6.0622 3 222 18.7 394.12
## lstat medv
## 1 4.98 24.0
## 2 9.14 21.6
## 3 4.03 34.7
## 4 2.94 33.4
## 5 5.33 36.2
## 6 5.21 28.7dim(Boston)## [1] 506 14Each row represent the set of predictor obeservations for a given Neighborhood in Boston. Each column represent each predictor variable for which an observation was made in 506 neighborhoods of Boston.
Question-b
Make some pairwise scatterplots of the predictors (columns) in this data set. Describe your findings:
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...Boston$chas <- as.numeric(Boston$chas)
Boston$rad <- as.numeric(Boston$rad)
pairs(Boston)
Not much can be discerned other than the fact that some variables appear to be correlated. A correlation matrix would be more helpful and question-c gives us the opportunity to make one.
Question-c
Are any of the predictors associated with per capita crime rate? If so, explain the relationship.
## i j cor p
## 1 crim zn -0.20046922 5.506472e-06
## 2 crim indus 0.40658341 0.000000e+00
## 4 crim chas -0.05589158 2.094345e-01
## 7 crim nox 0.42097171 0.000000e+00
## 11 crim rm -0.21924670 6.346703e-07
## 16 crim age 0.35273425 2.220446e-16
## 22 crim dis -0.37967009 0.000000e+00
## 29 crim rad 0.62550515 0.000000e+00
## 37 crim tax 0.58276431 0.000000e+00
## 46 crim ptratio 0.28994558 2.942924e-11
## 56 crim black -0.38506394 0.000000e+00
## 67 crim lstat 0.45562148 0.000000e+00
## 79 crim medv -0.38830461 0.000000e+00Based on the correlation coefficients and their corresponding p-values, there is indeed an association between the per capita crime rate (crim) and the other predictors.
Question-d
Do any of the suburbs of Boston appear to have particularly high crime rates? Tax rates? Pupil-teacher ratios? Comment on the range of each predictor.
summary(Boston$crim)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.00632 0.08204 0.25650 3.61400 3.67700 88.98000summary(Boston$tax)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 187.0 279.0 330.0 408.2 666.0 711.0summary(Boston$ptratio)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 12.60 17.40 19.05 18.46 20.20 22.00library(ggplot2)
qplot(Boston$crim, binwidth=5 , xlab = "Crime rate", ylab="Number of Suburbs" )
qplot(Boston$tax, binwidth=50 , xlab = "Full-value property-tax rate per $10,000", ylab="Number of Suburbs")
qplot(Boston$ptratio, binwidth=5, xlab ="Pupil-teacher ratio by town", ylab="Number of Suburbs")
Considering that the median and maximum crime rate values are respectively about 0.26% and 89%, there are indeed some neighborhoods where the crime rate is alarmingly high
selection <- subset( Boston, crim > 10)
nrow(selection)/ nrow(Boston)## [1] 0.106719411% of the neighborhood’s have crime rates above 10%;
selection <- subset( Boston, crim > 50)
nrow(selection)/ nrow(Boston)## [1] 0.0079051380.8% of the neighborhoods have crim rates above 50%
Based on the histogram of the Tax rates, they are few neighborhoods where rates are relative higher. The median and average tax amount are $330 and $408.20 ( per Full-value property-tax rate per $10,000) respectively.
selection <- subset( Boston, tax< 600)
nrow(selection)/ nrow(Boston)## [1] 0.72924973% of the neighborhood pay under $600
selection <- subset( Boston, tax> 600)
nrow(selection)/ nrow(Boston)## [1] 0.27075127% of the neighborhood pay over $600
Question-e
How many of the suburbs in this data set bound the Charles river?
nrow(subset(Boston, chas ==1)) ## [1] 35There are 35 such suburbs.
Question-f
What is the median pupil-teacher ratio among the towns in this data set?
summary(Boston$ptratio)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 12.60 17.40 19.05 18.46 20.20 22.00The median pupil-teacher ratio is 19 pupils for each teacher
Question-g
Which suburb of Boston has lowest median value of owner occupied homes?What are the values of the other predictors for that suburb, and how do those values compare to the overall ranges for those predictors?
selection <- Boston[order(Boston$medv),]
selection[1,]## crim zn indus chas nox rm age dis rad tax ptratio black
## 399 38.3518 0 18.1 0 0.693 5.453 100 1.4896 24 666 20.2 396.9
## lstat medv
## 399 30.59 5Suburb #399 with a median value of $5000. We can use the following summary information to answer part-2 of this question
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00Suburd #399: * Crime is very high compared to median and average rates of all Boston neighborhoods. * No residential land zoned for lots over 25,000 sq.ft. This applies to more than half of the neighborhoods in Boston * Proportion of non-retail business acres per town is very high compared to most suburbs. * This suburd is not one of the suburbs that bound the Charles river. * Nitrogen oxides concentration (parts per 10 million) is one of the highest. * Average number of rooms per dwelling is one of the lowest * Highest proportion of owner proportion of owner-occupied units built prior to 1940. * One of the lowest weighted mean of distances to five Boston employment centres. * Highest index of accessibility to radial highways. * One of the highest full-value property-tax rate per $10,000. * One of the highest pupil-teacher ratio by town * Highest value for 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town. * One of the highest lower status of the population (percent) * Lowest median value of owner-occupied homes in $1000s.
Based on the list above, suburb 399 can be classified as one of the least desirable places to live in Boston.
Question-h
In this data set, how many of the suburbs average more than seven rooms per dwelling? More than eight rooms per dwelling? Comment on the suburbs that average more than eight rooms per dwelling.
rm_over_7 <- subset(Boston, rm>7)
nrow(rm_over_7) ## [1] 64rm("rm_over_7")There are 64 suburbs with more than 7 rooms per dwelling.
rm_over_8 <- subset(Boston, rm>8)
nrow(rm_over_8) ## [1] 13There are 13 suburbs with more than 7 rooms per dwelling
summary(rm_over_8)## crim zn indus chas
## Min. :0.02009 Min. : 0.00 Min. : 2.680 Min. :0.0000
## 1st Qu.:0.33147 1st Qu.: 0.00 1st Qu.: 3.970 1st Qu.:0.0000
## Median :0.52014 Median : 0.00 Median : 6.200 Median :0.0000
## Mean :0.71879 Mean :13.62 Mean : 7.078 Mean :0.1538
## 3rd Qu.:0.57834 3rd Qu.:20.00 3rd Qu.: 6.200 3rd Qu.:0.0000
## Max. :3.47428 Max. :95.00 Max. :19.580 Max. :1.0000
## nox rm age dis
## Min. :0.4161 Min. :8.034 Min. : 8.40 Min. :1.801
## 1st Qu.:0.5040 1st Qu.:8.247 1st Qu.:70.40 1st Qu.:2.288
## Median :0.5070 Median :8.297 Median :78.30 Median :2.894
## Mean :0.5392 Mean :8.349 Mean :71.54 Mean :3.430
## 3rd Qu.:0.6050 3rd Qu.:8.398 3rd Qu.:86.50 3rd Qu.:3.652
## Max. :0.7180 Max. :8.780 Max. :93.90 Max. :8.907
## rad tax ptratio black
## Min. : 2.000 Min. :224.0 Min. :13.00 Min. :354.6
## 1st Qu.: 5.000 1st Qu.:264.0 1st Qu.:14.70 1st Qu.:384.5
## Median : 7.000 Median :307.0 Median :17.40 Median :386.9
## Mean : 7.462 Mean :325.1 Mean :16.36 Mean :385.2
## 3rd Qu.: 8.000 3rd Qu.:307.0 3rd Qu.:17.40 3rd Qu.:389.7
## Max. :24.000 Max. :666.0 Max. :20.20 Max. :396.9
## lstat medv
## Min. :2.47 Min. :21.9
## 1st Qu.:3.32 1st Qu.:41.7
## Median :4.14 Median :48.3
## Mean :4.31 Mean :44.2
## 3rd Qu.:5.12 3rd Qu.:50.0
## Max. :7.44 Max. :50.0