Lab 1 - Chapter 2

Question 9

a) Which of the predictors are quantitative, and which are qualitative?
#we first load the Auto dataset
library(ISLR2)
## Warning: package 'ISLR2' was built under R version 4.4.2
#inspect the first five elements in the dataset
head(Auto, 5)

Response: Quantitative predictors are mpg, cylinders, displacement, horsepower, weight, acceleration, year. Qualitative data would be origin, and name.

b) What is the range of each quantitative predictor? You can answer this using the range() function
#first we ensure we remove missing values from the data
anyNA(Auto)
## [1] FALSE
# we don't seem to have missing values in this dataset

#evaluating the range for quantitative predictors
cat("mpg range: ", range(Auto$mpg), "\n") 
## mpg range:  9 46.6
cat("cylinders range: ", range(Auto$cylinders), "\n") 
## cylinders range:  3 8
cat("Displacement range: ", range(Auto$displacement), "\n") 
## Displacement range:  68 455
cat("Horsepower range: ", range(Auto$horsepower), "\n") 
## Horsepower range:  46 230
cat("Weight range: ", range(Auto$weight), "\n") 
## Weight range:  1613 5140
cat("Acceleration range: ", range(Auto$acceleration), "\n") 
## Acceleration range:  8 24.8
cat("Year range: ", range(Auto$year), "\n") 
## Year range:  70 82

Response: The above are the ranges for the various quantitative predictors.

c) What is the mean and standard deviation of each quantitative predictor?
#To evaluate the mean for all the quantitative predictors

#quantitative columns to exclude
quantitative_cols <- c("origin", "name")
cat("Mean of quantitative predictors\n")
## Mean of quantitative predictors
sapply(Auto[, !names(Auto) %in% quantitative_cols], mean, na.rm=TRUE)
##          mpg    cylinders displacement   horsepower       weight acceleration 
##    23.445918     5.471939   194.411990   104.469388  2977.584184    15.541327 
##         year 
##    75.979592
#To evaluate the standard deviation for all the quantitative predictors

cat("\n\nStandard Deviations for quantitative predictors\n")
## 
## 
## Standard Deviations for quantitative predictors
sapply(Auto[, !names(Auto) %in% quantitative_cols], sd, na.rm=TRUE)
##          mpg    cylinders displacement   horsepower       weight acceleration 
##     7.805007     1.705783   104.644004    38.491160   849.402560     2.758864 
##         year 
##     3.683737

Response: The standard deviations and means of the quantitative predictors is as shown above.

d) Now remove the 10th through 85th observations. What is the range, mean, and standard deviation of each predictor in the subset of the data that remains?
#removing the 10th to 85th observations
Auto_subset <- Auto[-(10:85),]

#evaluating the mean and sd

#quantitative columns to exclude
quantitative_cols <- c("origin", "name")
cat("Mean of quantitative predictors\n")
## Mean of quantitative predictors
sapply(Auto[, !names(Auto_subset) %in% quantitative_cols], mean, na.rm=TRUE)
##          mpg    cylinders displacement   horsepower       weight acceleration 
##    23.445918     5.471939   194.411990   104.469388  2977.584184    15.541327 
##         year 
##    75.979592
#To evaluate the standard deviation for all the quantitative predictors

cat("\n\nStandard Deviations for quantitative predictors\n")
## 
## 
## Standard Deviations for quantitative predictors
sapply(Auto[, !names(Auto_subset) %in% quantitative_cols], sd, na.rm=TRUE)
##          mpg    cylinders displacement   horsepower       weight acceleration 
##     7.805007     1.705783   104.644004    38.491160   849.402560     2.758864 
##         year 
##     3.683737
e) Using the full data set, investigate the predictors graphically, using scatterplots or other tools of your choice. Create some plots highlighting the relationships among the predictors. Comment on your findings.
#Horsepower vs Miles Per Gallon relationship
plot(Auto$horsepower, Auto$mpg,
     col="blue",
     pch=19,
     xlab = "Miles Per Gallon",
     ylab = "Horsepower",
     main="Horsepower vs MPG")

  • For miles per gallon vs horsepower relationship, we see that horsepower generally tend to reduce with increase in miles per gallon. This makes sense since in actual sense vehicles designed to have better fuel economy, i.e. higher MPG, often have smaller engines.
#we investigate the relationship between weight of car and horsepower
plot(
  Auto$horsepower,
  Auto$weight,
  pch=19,
  col="blue",
  xlab = "Weight",
  ylab="Horsepower",
  main = "Horsepower vs. Weight"
)

  • From the scatter plot above, we see that the Horsepower generally tends to increase with weight. This makes sense since for heavier vehicles, we would expect them to have more powerful engine.
f) Suppose that we wish to predict gas mileage (mpg) on the basis of the other variables. Do your plots suggest that any of the other variables might be useful in predicting mpg? Justify your answer.
  • Response: To predict gas mileage, I could use weight, horsepower, and acceleration. These three seems to influence the miles per gallon in a real world set up.
#mpg vs. weight
plot(
  Auto$mpg,
  Auto$weight,
  pch=19,
  col="blue",
  xlab = "Weight",
  ylab="mpg",
  main = "mpg vs. Weight"
)
lines(lowess(Auto$mpg,Auto$weight), col="red", lwd=2)

#mpg vs. acceleration
plot(
  Auto$mpg,
  Auto$acceleration,
  pch=19,
  col="blue",
  xlab = "acceleration",
  ylab="mpg",
  main = "mpg vs. acceleration"
)
lines(lowess(Auto$mpg,Auto$acceleration), col="red", lwd=2)

#mpg vs. horsepower
plot(
  Auto$mpg,
  Auto$horsepower,
  pch=19,
  col="blue",
  xlab = "horsepower",
  ylab="mpg",
  main = "mpg vs. horsepower"
)
lines(lowess(Auto$mpg,Auto$horsepower), col="red", lwd=2)

Question 10

a) How many rows are in this data set? How many columns? What do the rows and columns represent?
#reading the dataset documentation
#?Boston
  • Response: The data has 506 rows and 13 columns. The rows represent housing units in 506 suburbs of Boston.
b) Make some pairwise scatterplots of the predictors (columns) in this data set. Describe your findings.
#crime vs. housing value
boston_ds <- Boston

plot(
  boston_ds$crim,
  boston_ds$medv,
  pch = 19,
  col = "blue",
  xlab = "Median Home Value",
  ylab = "Per Capital Crime Rate"
)
lines(lowess(boston_ds$crim,boston_ds$medv), col="red", lwd=2)

  • We see the bulk of the data being in low median homes, and this has an effect on crime, where we notice a higher per capital rate by town in low median value homes. Often, you will find areas with lower median home values to be associated with a higher crime rate.
#Nitrogen Oxides concentration vs. proportion of non-retail business areas

boston_ds <- Boston

plot(
  boston_ds$nox,
  boston_ds$indus,
  pch = 19,
  col = "blue",
  xlab = "industrial ares proportion per town",
  ylab = "Nitrogen Oxides concentrations"
)
lines(lowess(boston_ds$nox,boston_ds$indus), col="red", lwd=2)

  • We see that as the proportion of industrial business acres per town increases, so does the concentrations of nitrogen oxide. This makes sense more industrial businesses per town would definitely tend to pollute more while keeping the other factors constant.
c) Are any of the predictors associated with per capita crime rate? If so, explain the relationship.
  • Response: Yes, as we have already seen in the previous plot, areas with lower median home values tend to be associated with higher crime rates.
d) Do any of the census tracts of Boston appear to have particularly high crime rates? Tax rates? Pupil-teacher ratios? Comment on the range of each predictor.
#evaluating the range of each of the predictor
#tax rate, crime rate, and pupil-teacher ratio

cat("Crime Rate Range:\t")
## Crime Rate Range:    
range(boston_ds$crim)
## [1]  0.00632 88.97620
cat("Tax Rate Range:\t")
## Tax Rate Range:  
range(boston_ds$tax)
## [1] 187 711
cat("Pupil-teacher ratio Range:\t")
## Pupil-teacher ratio Range:   
range(boston_ds$ptratio)
## [1] 12.6 22.0

  • For crime rate, the value ranges from 0.00632 to 88.97620 suggesting that there are areas with very low crime rate, and others with extremely high crime rate.

  • For tax rate, the value ranges from 187 to 711, indicating also a large disparity in tax rate in Boston homes.

  • For pupil-teacher ration, the value ranges from 12.5 to 22.0. The difference is quite glaring i.e. 12 students per teacher vs. 22 students per teacher. This means that some neighborhoods in Boston have crowded classrooms, which could imply that students don’t get teacher support as much,

  • Overall, for the census tracts that we have considered above, Boston homes exhibit a significant variation across those census tracts.

e) How many of the census tracts in this data set bound the Charles river?
# for tract that borders Charles River, value =  1, otherwise value = 0
cat("Census Tracts that bound by Charles River:\t")
## Census Tracts that bound by Charles River:   
sum(boston_ds$chas)
## [1] 35
  • Therefore, we have 35 census tracts that are bounded by Charles River.
f) What is the median pupil-teacher ratio among the towns in this data set?
#median pupil-teacher ratio
cat("Medial pupil-teacher ratio:\t")
## Medial pupil-teacher ratio:  
median(boston_ds$ptratio)
## [1] 19.05
  • The median value for pupil-teacher ratio is 19. This means that half of the census tracts have a pupil-teacher ratio below 19. From the median we can see that the pupil teacher ratio is being significantly skewed by outliers, when you look at the range of pupil-teacher ratio (12.6, 22.0).
g) Which census tract of Boston has lowest median value of owner occupied homes? What are the values of the other predictors for that census tract, and how do those values compare to the overall ranges for those predictors? Comment on your findings.
#census tract with lowest median value
min_medv_index <- which.min(boston_ds$medv)
boston_ds[min_medv_index, ]

  • The above census tract has the lowest median value of owner occupied homes. This census tract has tax rate of 666 which is near the maximum in the range (187, 711). Ideally, you would expect neighborhoods in low median values to have lower tax rates.

  • For the pupil teacher ratio, the census tract has a value of 20.2, again near the maximum range of (12.6, 22.0). This indicates a scenario of perhaps crowded classrooms.

  • Lastly for crime rate, this census tract has a relatively high crime rate of 38% in a range of (0.00632, 88.97620). This was too expected, since neighborhoods with low median values tend to be associated with a higher crime rate.

h) In this data set, how many of the census tracts average more than seven rooms per dwelling? More than eight rooms per dwelling? Comment on the census tracts that average more than eight rooms per dwelling.
#no of census tracts with more than 7 rooms
cat("number of census tracts with more than 7 rooms:\t")
## number of census tracts with more than 7 rooms:  
sum(boston_ds$rm > 7)
## [1] 64
#no of census tracts with more than 8 rooms
cat("\nnumber of census tracts with more than 8 rooms:\t")
## 
## number of census tracts with more than 8 rooms:  
sum(boston_ds$rm > 8)
## [1] 13
  • 64 census tracts have an average of 7 rooms per dwelling, and only 13 census tracts have average of 8 rooms per dwelling. This means that the number of rooms per dwelling varies across different census tracts, and only a small proportion of the tracts have very large homes.