General Guidelines
mixed1 <- c(TRUE, FALSE, FALSE, TRUE, 4, 0, 3) mixed1
Homework 2
Stats 20 Lec 1 and 2 Fall 2020
Please use R Markdown for your submission. Include the following files: • Your .Rmd file.
• The compiled/knitted HTML document.
• Your .bib file (if needed).
Name your .Rmd file with the convention 123456789_stats20_hw0.Rmd, where 123456789 is replaced with your UID and hw0 is updated to the actual homework number. Include your first and last name and UID in your exam as well. When you knit to HTML, the HTML file will inherit the same naming convention.
The knitted document should be clear, well-formatted, and contain all relevant R code, output, and explanations. R code style should follow the Tidyverse style guide: https://style.tidyverse.org/.
Note: All questions on this homework should be done using only functions or syntax discussed in Chapters 1–2 of the lecture notes. No credit will be given for use of outside functions.
Basic Questions
Collaboration on basic questions must adhere to Level 0 collaboration described in the Stats 20 Collaboration Policy.
The following code is used in Questions 1 and 2.
Consider the following code:
[1] 1 0 0 1 4 0 3
[1] "TRUE" "FALSE" "FALSE" "TRUE" "4" "0" "3"
[1] "1" "0" "0" "1" "4" "0" "3"
mixed2 <- c(TRUE, FALSE, FALSE, TRUE, 4, 0, "3") mixed2
mixed3 <- c(c(TRUE, FALSE, FALSE, TRUE, 4, 0), "3") mixed3
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Question 1
The objective of this question is to help further your understanding of the mode hierarchy and the order in which commands are evaluated.
(a)
Explain why mixed2 and mixed3 produce different results. (b)
Use the same input values of TRUE, FALSE, FALSE, TRUE, 4, 0, "3" with the c() function to create the following vector:
[1] "TRUE" "FALSE" "0" "1" "4" "0" "3"
Question 2
The objective of this question is to introduce the concept of type casting.
Type Casting: The as.logical(), as.numeric(), and as.character() functions allow us to coerce (or
cast) a vector into one of a different mode. For example:
as.logical(c("TRUE", "FALSE", "TRUE")) [1] TRUE FALSE TRUE
as.numeric(c("4", "0", "3")) [1] 4 0 3
as.character(c(TRUE, FALSE, TRUE)) [1] "TRUE" "FALSE" "TRUE"
(a)
Explain why as.numeric(mixed2) and as.numeric(mixed3) produce different results. (b)
Explain why as.logical(mixed2) and as.logical(mixed3) produce different results. (c)
Use type casting functions to coerce mixed3 into a meaningful logical vector (i.e., with no NA values).
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Question 3
The objective of this question is to give practice with the seq() and rep() functions and understand their use and limitations.
(a)
Generate the following sequence in four ways: using (i) the seq() function, (ii) the colon : operator, (iii) the seq_len() function, and (iv) the seq_along() function.
[1] 1 2 3 4 5 6 7 8 910111213141516171819202122232425
(b)
Using seq() and rep() as needed, create the following vectors in R.
Note: Sequences should be written in such a way that they would be generalizable if the desired pattern was longer. For example, c(1, 2, 3, 4, 5) should be written as seq(1, 5), and c(1, 1, 1, 1, 1) should be written as rep(1, 5). Extending these patterns to 1000 values would be prohibitively cumbersome without using seq(1, 1000) and rep(1, 1000).
(i)
[1] 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0
(ii)
[1] -1 0 3 5 -1 0 3 5 -1 0 3 5 -1 0 3 5 -1 0 3 5
(iii)
[1] 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1
(iv)
[1] 5 5 5 5 5 4 4 4 4 4 3 3 3 3 3 2 2 2 2 2 1 1 1 1 1
(v)
[1] 10 9 9 8 8 8 7 7 7 7 6 6 6 6 6
(c)
Explain why the following sequence cannot be generated using only the seq() and rep() functions in the same way as the sequences in (a). What makes this pattern different from the patterns in (a)? How can you generate this sequence?
[1] 10 9 8 7 6 9 8 7 6 5 8 7 6 5 4 7 6 5 4 3 6 5 4 3 2
Hint: Since this sequence cannot be generated using only seq() and rep(), consider another operation in
addition to seq() and rep() to combine simpler sequences into this more complicated pattern. Question 4
R has built-in trigonometric functions, such as sin(), cos(), tan(), asin(), acos(), and atan(). The sin() function computes the sine of any numeric input.
Using at most two lines of code, compute sin(kπ), for k = 0, 1, 2, . . . , 100. Explain your result. Does the output match what you would expect mathematically?
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Question 5
The objective of this question is to give practice with vectorized operations and writing functions using vectors.
The coefficient of skewness measures how much the distribution of a sample differs from symmetry. A perfectly symmetric distribution will have a skewness of 0. If the skewness coefficient is significantly greater than 0, then the distribution is right-skewed. If the skewness coefficient is significantly less than 0, then the distribution is left-skewed.
The skewness coefficient for a sample of values x1, x2, . . . , xn is defined as
skew =
√n 3 n (xi−x ̄)
i=1 , n 3/2
( x i − x ̄ ) 2 i=1
1 n
where x ̄ = n (a)
xi is the sample mean.
i=1
Write a function called my_skew() that computes the skewness for any numeric input vector x. (b)
Use your my_skew() function to compute the skewness of Chris Traeger’s running times from the lecture notes. Interpret your answer.
running_times <- c(51, 40, 57, 34, 47, 50, 50, 56, 41, 38)
(c)
Through algebraic manipulation, a one-pass formula for the coefficient of skewness is
skew =
√n3 n2 3 n xi−3x ̄ xi+2nx ̄
i=1 i=1 . n 3/2
x 2i − n x ̄ 2 i=1
Write a function called my_skew_one() that computes the skewness for any numeric input vector x using the one-pass formula. Verify that your new function gives the same answer as your function from (a).
(d)
Multiply the running times by 1010 (1e10) and compute the skewness on the new vector using both of your skewness functions. Do your results differ from your answers in (b) and (c)? Explain intuitively or mathematically why or why not.
(e)
Even though the skewness of Chris Traeger’s running times should appear to be the same value in (b) and (c), you likely will find the following result when considering the difference in the values:
my_skew(running_times) - my_skew_one(running_times)
[1] -4.157785e-14
Explain why the difference is not identically 0.
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Intermediate Questions
Collaboration on intermediate questions must adhere to Level 1 collaboration described in the Stats 20 Collaboration Policy.
Several exercises throughout this course will ask you to code your own version of basic built-in functions from scratch. While you will likely not need to rewrite existing functions from scratch when using R in practice, the thought process of thinking through and writing these functions allows you to build your skills in breaking down complicated problems into simpler steps and more deeply understanding the fundamental toolkit you are building throughout the course.
A general strategy for how to think through writing functions from scratch:
1. Come up with small general examples and think about what your function ideally should do for those examples. Do you notice patterns or using similar logic for every example?
2. Formalize and generalize the logic you used in your small examples to the general case. How should your function work on the intended input argument(s) in general? Outline or describe the steps your function needs to do to output the desired result.
3. Consider any edge cases, i.e., valid inputs that your function is supposed to work on but may not use the same logic as the general case to work. See if the logic you used in general extends to the edge case. If not, think about whether the general case can be modified to accommodate the edge case.
In Chapter 4, we will consider a flow control statement (an if or if-else statement) that allows you to use/write entirely separate logic for different cases.
Question 6
(a)
Write a function called my_length() that computes the number of elements of a vector without the length() function. The output of my_length(x) and length(x) should be identical for any vector x. Your function must work for vectors of any mode and vectors of length 0.
(b)
Test your my_length() function from (a) with the following inputs: (i) numeric(0)
(ii) seq(1, 99, by = 2)
(iii) c("friends", "waffles", "work")
Question 7
(a)
Write a function called my_rev() that reverses the order of elements in a vector without the rev() function. The output of my_rev(x) and rev(x) should be identical for any vector x. Your function must work for vectors of length 0.
(b)
Test your my_rev() function from (a) with the following inputs: (i) numeric(0)
(ii) seq(1, 99, by = 2)
(iii) c("friends", "waffles", "work")
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Question 8
(a)
Write a function called front() that inputs a vector x and a number n and returns the first n elements of x. When n is greater than length(x), your function should just return x.
With n = 6, verify that your front() function works on the vectors numeric(0), seq(1, 99, by = 2), and c("friends", "waffles", "work").
(b)
Write a function called back() that inputs a vector x and a number n and returns the last n elements of x. When n is greater than length(x), your function should just return x.
With n = 6, verify that your back() function works on the vectors numeric(0), seq(1, 99, by = 2), and c("friends", "waffles", "work").
(c)
Read through the following pseudocode:
FUNCTION: insert_at()
INPUTS: x, the vector the values are to be added to.
values, the values to be included in the modified vector.
at, the starting index of the inserted values.
OUTPUT: a vector with values inserted into vector x at the specified position.
beginning <- subset of `x` values with indices less than `at`
ending <- subset of `x` values with indices greater-than or equal to `at`
new_vec <- combination of beginning, values, and ending
RETURN: new_vec
Use your front() and/or back() functions from (a) and (b) to write a function called insert_at() that inserts elements in a vector.
(d)
Verify that your insert_at() function works by executing the following commands: insert_at(1:5, c(0, 0, 0), at = 1)
[1] 0 0 0 1 2 3 4 5
insert_at(1:5, c(0, 0, 0), at = 3) [1] 1 2 0 0 0 3 4 5
insert_at(1:5, c(0, 0, 0), at = 5) [1] 1 2 3 4 0 0 0 5
insert_at(1:5, c(0, 0, 0), at = 6) [1] 1 2 3 4 5 0 0 0
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