Intro to R Part 2

# Home-runs so far
HR_before <- c(11, 13, 12)
# Average Number of Home-runs per season wanted
wanted_HR <- 20
# Number of seasons
n_seasons <- 4
# Needed Home-runs on season 4
x_4 <- n_seasons*wanted_HR - sum(HR_before)
# Minimum number of Home-runs needed by Robert
x_4

[1] 44

According to the calculations above, Robert must hit 44 home-runs or better on this season to get an average number of home-runs per season of at least 20.

# Robert's performance
Robert_HRs <- c(11, 13, 12,44)
# Find mean
mean(Robert_HRs)

[1] 20

Robert 44 home runs in the fourth season results in an overall average of 20 home runs per season.

# Robert's actual performance
Robert_HR <-c(11, 13, 12, 38)
mean(Roberts_HR)

[1] 18.5

Robert average 18.5 per season.

sd(Robert_HR)

[1] 13.02562

Robert’s home runs have a standard deviation of approximately 13.35 home runs.

max(Robert_HR)

[1] 38

Robert’s highest number of home runs in a single season was 38.

# Find the minimum number of home-runs during the four seasons period
min(Robert_HR)

[1] 11

This means Robert’s lowest number of home runs in a single season was 11.

summary(Robert_HR)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  11.00   11.75   12.50   18.50   19.25   38.00 

Robert’s actual four-season performance ranges from 11 to 38 home runs, with an average of 18.5 home runs per season.

# Walks so far
Walks_before <- c(79, 108, 41, 145, 135)

# Average Number of Walks per season wanted
wanted_Walks <- 100

# Number of seasons
n_seasons <- 6

# Needed Walks on season 6
x_6 <- n_seasons*wanted_Walks - sum(Walks_before)

# Minimum number of Walks needed by Soto
x_6

[1] 92

Question 1 Juan Soto needs 92 walks in his sixth season to have an overall average of 100 walks per season.

n_1 <- 10
n_2 <- 4
y_1 <- 72000
y_2 <- 84000
# Mean salary overall
salary_ave <-  (n_1*y_1 + n_2*y_2)/(n_1+n_2)
salary_ave

[1] 75428.57

The overall average salary is approximately $75,428.57.

n_1 <- 7
n_2 <- 9

y_1 <- 102000
y_2 <- 91000

salary_ave <- (n_1*y_1 + n_2*y_2)/(n_1+n_2)

salary_ave

[1] 95812.5

Question 2 The mean salary of all 16 professional players is $95,812.50 per week.

contract_length <- read.table("allcontracts.csv", header = TRUE, sep = ",")
contract_years <- contract_length$years

Importing the contract data and extracting the years column.

# Mean 
contracts_mean  <- mean(contract_years)
contracts_mean

[1] 3.458918

The mean average contract length from the contract_years.

# Median
contracts_median <- median(contract_years)
contracts_median

[1] 3

The median contract length from the contract_years data.

# Find number of observations
contracts_n <- length(contract_years)
# Find standard deviation
contracts_sd <- sd(contract_years)

Finds two important descriptive statistics for the contract data

contracts_w1sd <- sum((contract_years - contracts_mean)/contracts_sd < 1)/ contracts_n
# Percentage of observation within one standard deviation of the mean
contracts_w1sd

[1] 0.8416834

Calculates the percentage of contract observations within one standard deviation below the mean.

## Within 2 sd
contracts_w2sd <- sum((contract_years - contracts_mean)/ contracts_sd < 2)/contracts_n
contracts_w2sd

[1] 1

This follows the empirical rule, where about 95% of observations in a normal distribution fall within 2 standard deviations of the mean.

## Difference from empirical 
contracts_w2sd - 0.95

[1] 0.05

## Within 3 sd 
contracts_w3sd <- sum((contract_years - contracts_mean)/ contracts_sd < 3)/contracts_n
contracts_w3sd

[1] 1

Code calculates the proportion of contract observations within 3 standard deviations below the mean. The data is 5 percentage points below the empirical rule expectation.

## Difference from empirical 
contracts_w3sd - 0.9973

[1] 0.0027

The data is 1.73 percentage points below the empirical rule expectation.

# Create histogram
hist(contract_years,xlab = "Years Left in Contract",col = "green",border = "red", xlim = c(0,8), ylim = c(0,225),
   breaks = 3)

Creates a histogram to visualize the distribution of contract lengths.