Homework 1

Question 1

a)

Since CEO salary is a continuous variable and we are modeling it as a function of the predictors (industry, number of employees, and profits), it’s a regression problem.

We are interested in understanding which factors have an impact on CEO salary. This means we are focused on interpreting the relationships between the predictors (industry, number of employees, and profits) and the response (CEO salary), rather than predicting CEO salaries for new firms. Thus, the goal here is inference—we want to identify and understand the importance of each factor and how it influences CEO salaries.

• n: 500, which is the number of firms in the dataset, representing the total number of observations.
  
• p: 3, as the predictors are industry (categorical), number of employees (numeric), and profits (numeric).

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b)

Since the response variable is binary (success or failure), it’s a classification problem.

We aim to predict whether a new product will be a success or a failure based on historical data about similar products. The predictors include price charged, marketing budget, competition price, and ten other variables. Here, the goal is prediction, as we are not focused on understanding the relationships between predictors and the response but rather on accurately forecasting the outcome (success or failure) for a new product.

• n: 20, which is the number of similar products in the dataset, representing the total number of observations.
  
• p: 13, as the predictors include price charged, marketing budget, competition price, and ten other variables, giving us a total of 13 predictors.

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c)

Since the percent change in the USD/Euro exchange rate in relation to the weekly changes in the world stock markets is the response variable, and it’s continuous, this is a regression problem.

The objective is to predict the future percentage change in the USD/Euro exchange rate based on weekly changes in stock market indices for the US, British, and German markets. This makes the goal prediction, as we are focused on using the relationships between the predictors and the response to forecast future changes in the exchange rate, not to understand or interpret the relationships themselves.

• n: 52, which is the number of weeks in 2012, representing the total number of observations.
  
• p: 3, as the predictors are the percentage changes in the US stock market, British stock market, and German stock market.

Question 2:

a)

# Please change path if checking this code on a seperate device
college <- read.csv("/Applications/R_Folder/College.csv") 

View(college)

b)

rownames(college) <- college[, 1]

View(college)

college <- college[, -1]

View(college)

c)

i)

summary(college)

   Private               Apps           Accept          Enroll    
 Length:777         Min.   :   81   Min.   :   72   Min.   :  35  
 Class :character   1st Qu.:  776   1st Qu.:  604   1st Qu.: 242  
 Mode  :character   Median : 1558   Median : 1110   Median : 434  
                    Mean   : 3002   Mean   : 2019   Mean   : 780  
                    3rd Qu.: 3624   3rd Qu.: 2424   3rd Qu.: 902  
                    Max.   :48094   Max.   :26330   Max.   :6392  
   Top10perc       Top25perc      F.Undergrad     P.Undergrad     
 Min.   : 1.00   Min.   :  9.0   Min.   :  139   Min.   :    1.0  
 1st Qu.:15.00   1st Qu.: 41.0   1st Qu.:  992   1st Qu.:   95.0  
 Median :23.00   Median : 54.0   Median : 1707   Median :  353.0  
 Mean   :27.56   Mean   : 55.8   Mean   : 3700   Mean   :  855.3  
 3rd Qu.:35.00   3rd Qu.: 69.0   3rd Qu.: 4005   3rd Qu.:  967.0  
 Max.   :96.00   Max.   :100.0   Max.   :31643   Max.   :21836.0  
    Outstate       Room.Board       Books           Personal   
 Min.   : 2340   Min.   :1780   Min.   :  96.0   Min.   : 250  
 1st Qu.: 7320   1st Qu.:3597   1st Qu.: 470.0   1st Qu.: 850  
 Median : 9990   Median :4200   Median : 500.0   Median :1200  
 Mean   :10441   Mean   :4358   Mean   : 549.4   Mean   :1341  
 3rd Qu.:12925   3rd Qu.:5050   3rd Qu.: 600.0   3rd Qu.:1700  
 Max.   :21700   Max.   :8124   Max.   :2340.0   Max.   :6800  
      PhD            Terminal       S.F.Ratio      perc.alumni   
 Min.   :  8.00   Min.   : 24.0   Min.   : 2.50   Min.   : 0.00  
 1st Qu.: 62.00   1st Qu.: 71.0   1st Qu.:11.50   1st Qu.:13.00  
 Median : 75.00   Median : 82.0   Median :13.60   Median :21.00  
 Mean   : 72.66   Mean   : 79.7   Mean   :14.09   Mean   :22.74  
 3rd Qu.: 85.00   3rd Qu.: 92.0   3rd Qu.:16.50   3rd Qu.:31.00  
 Max.   :103.00   Max.   :100.0   Max.   :39.80   Max.   :64.00  
     Expend        Grad.Rate     
 Min.   : 3186   Min.   : 10.00  
 1st Qu.: 6751   1st Qu.: 53.00  
 Median : 8377   Median : 65.00  
 Mean   : 9660   Mean   : 65.46  
 3rd Qu.:10830   3rd Qu.: 78.00  
 Max.   :56233   Max.   :118.00  

ii)

college$Private <- ifelse(college$Private == "Yes", 1, 0)

pairs(college[,1:10], main = "Pairwise Scatterplots of the First Ten Variables")

iii)

boxplot(Outstate ~ Private, data = college,
        main = "Out-of-State Tuition for Private and Public Universities",
        xlab = "University Type (Private/Public)", ylab = "Out-of-State Tuition",
        col = c("lightblue", "lightgreen"))

iv)

Elite <- rep("No", nrow(college))
Elite[college$Top10perc > 50] <- "Yes"
Elite <- as.factor(Elite)
college <- data.frame(college, Elite)

summary(college$Elite)

 No Yes 
699  78 

boxplot(Outstate ~ Elite, data = college,
        main = "Outstate Tuition vs Elite Status",
        xlab = "Elite University (yes/no)", ylab = "Outstate Tuition",
        col = c("pink", "purple"))

v)

par(mfrow = c(2, 2))


hist(college$Apps, main = "Histogram of Applications", xlab = "Number of Applications", col = "lightblue", breaks = 20)
hist(college$Accept, main = "Histogram of Acceptances", xlab = "Number of Acceptances", col = "lightgreen", breaks = 30)

hist(college$Enroll, main = "Histogram of Enrollments", xlab = "Number of Enrollments", col = "lightpink", breaks = 15)
hist(college$Outstate, main = "Histogram of Outstate Tuition", xlab = "Outstate Tuition", col = "lightgray", breaks = 25)

vi)

summary(college[, c("Apps", "Accept", "Enroll", "Outstate")])

      Apps           Accept          Enroll        Outstate    
 Min.   :   81   Min.   :   72   Min.   :  35   Min.   : 2340  
 1st Qu.:  776   1st Qu.:  604   1st Qu.: 242   1st Qu.: 7320  
 Median : 1558   Median : 1110   Median : 434   Median : 9990  
 Mean   : 3002   Mean   : 2019   Mean   : 780   Mean   :10441  
 3rd Qu.: 3624   3rd Qu.: 2424   3rd Qu.: 902   3rd Qu.:12925  
 Max.   :48094   Max.   :26330   Max.   :6392   Max.   :21700  

The dataset reveals significant variability across universities. Applications range from as low as 81 to as high as 48,094, with a median of 1,558. This indicates that while a few universities receive an exceptionally large number of applications, most fall far below these outliers. Acceptances and enrollments follow similar patterns, with medians of 1,110 and 434, respectively, but there are still schools admitting and enrolling far more students. Out-of-state tuition is more consistent, with a median of $9,990 and a range from $2,340 to $21,700. Most universities tend to cluster around $10,000 in tuition costs.

The histograms further illustrate these trends. Applications, acceptances, and enrollments are all heavily skewed, with a few universities standing out as extreme outliers. In contrast, out-of-state tuition is distributed more evenly. These findings show the diversity among universities, from smaller, lower-cost institutions to larger schools with higher tuition fees and more substantial student populations.