Homework 1 - Applied Statistical Computing

In introductory statistics, a metric called the standard deviation is often used to calculate the “spread” in the data. Formally, for a collection of data points, \(y_1, \dots, y_n\), the standard deviation is defined as:

\[SD(y_1, \dots, y_n) = \sqrt{\frac{1}{n}\sum_{i=1}^{n} (y_i - \bar{Y})^2}\]

where \(\bar{Y}\) is the average of values \(\bar{Y}=\frac{1}{n}\sum_{i=1}^n y_i\).

Q0

Please create a numeric vector called die that contains each of the integer values from 1 through 6 (inclusive) in the following ways (order does not matter). Please use print(die) after each method so we can see the output.

• Using the c() function (please ignore the fact that if you check the class() here you will see numeric instead of integer)
• Using the : operator after seeing how it behaves after running the following examples:
  • 4:10
  • 0:1
  • -3:3
• By modifying the code sample(k) where k is an integer. Please read the documentation in ?sample and see how sample() will behave when its first argument is an integer.

Q1

Please calculate the standard deviation of die without using the sd() function but by correcting the following code. You should not need to add lines of code to complete this.

die <- 
n <- length(die)
die_avg <- mean(die]
sum_sq_diff <- 0
for i in die {
    diff_from_avg <- die_avg - i
    sq_diff <- diff_from_avg^2
    sum_sq_diff <- sq_diff
}
theoretical_sd <- sqrt(sum_sq_diff)
print(theoretical_sd)

Q2

Please apply the sd() function on die from Q1 and use this information to verify if the sd() function is calculating the same standard deviation as we have defined above?

Q3

What happens when you forget to the closing parentases when calling a functionin the R console (i.e. the console section in Rstudio)? E.g. c(1, 2, 3 Please use words to describe the behavior (notice the symbol change in the prompt).

Q4

In class, we’ve shown that the sample average follows the law of large numbers, please verify if sd() seems to follow the law of large numbers using our dice roll example, i.e. does applying sd() on a larger samples of dice rolls converge towards the theoretical standard deviation of a die, Yes/No and please describe the output you see from your code that supports your answer?

Important: this is NOT a proof of any kind!

Side comment: in your introductory statistics course, you will learn that the statistic used based on the sample to “estimate” the population statistic is often slightly different.

Q5

The documentation on prob within ?sample is not entirely obvious. Please use a for-loop to simulate larger and larger sample sizes from the following piece of code: sample(c(1, 0), n, replace=TRUE, prob=c(3, 1))

then analyze the output to answer what does prob do? Please write the code and explain your answer. You should decide what sample sizes are reasonable.