An unidentified man drinking a non-alcoholic beer.

An unidentified man drinking a non-alcoholic beer.

Readings

This assignment is based on the following readings:

Assignment Goals

Examples

# Vector of specific values

c(37, 45, 23, 54, 66)   # Numeric
c("A", "B", "C", "D")   # Character

# Vector of integers 1 to 5

c(1, 2, 3, 4, 5)                       # using c()
1:5                                    # using a:b
seq(from = 1, to = 5, by = 1)          # using seq()
seq(from = 1, to = 5, length.out = 5)  # same as above using length.out

# Vector of multiples of 10 from 10 to 50

c(10, 20, 30, 40, 50)
seq(from = 10, to = 50, by = 10)
seq(from = 10, to = 50, length.out = 5)

# Assign vectors to objects

data_A <- c(37, 45, 23, 54, 66)
data_B <- seq(from = 1, to = 100, by = 2)

# Calculate descriptive statistics

mean(c(37, 45, 23, 54, 66))
mean(data_A)

median(seq(from = 1, to = 100, by = 2))
median(data_B)

# Vector arithmetic

a <- c(1, 2, 3, 4, 5)
a * 10   # Multiply all elements by 10
a + .5   # Add .5 to all elements

b <- c(10, 20, 30, 40, 50)
a + b    # Add a and b element-wise

# Generate samples from distributions

sample(1:10, size = 2, replace = FALSE) # 2 values from the integers from 1 to 10
rnorm(n = 10, mean = 50, sd = 2)        # 10 values from Normal(mean = 50, sd = 2)
rbinom(n = 100, size = 1, prob = .5)    # 100 coin flips (binomial with size = 1, prob = .5)

Get started

  1. Open RStudio. Open a new R script (File – New File – R Script), and save it as wpa_1_LastFirst.R (where Last and First is your last and first name). At the top of your script write the assignment number, your name and date (as comments!). For the rest of the assignment, when you answer a task, indicate which task you are answering with appropriate comments as follows:

Here is an example of how your wpa_1_LastFirst.R file could look

# Assignment: WPA X
# Name: LAST, FIRST
# Date: DAY MONTH YEAR


# TASK 1
1 + 1

# TASK 2
2 + 2

# ...

Does drinking non-alcoholic beer affect cognitive performance?

A psychologist has a theory that some of the negative cognitive effects of alcohol are the result of psychological rather than physiological processes. To test this, she has 12 participants perform a cognitive test before and after drinking non-alcoholic beer which was labelled to contain 5% alcohol. Results from the study, including some demographic data, are presented in the following table. Note that higher scores on the test indicate better performance.

id before after age sex eye_color
1 45 43 20 male blue
2 49 50 19 female blue
3 40 61 22 male brown
4 48 44 20 female brown
5 44 45 27 male blue
6 70 20 22 female blue
7 90 85 22 male brown
8 75 65 20 female brown
9 80 72 25 male blue
10 65 65 22 female blue
11 80 70 24 male brown
12 52 75 22 female brown

Creating vectors from scratch

We’ll start by creating vector objects representing each vector of data (i.e.; column from the table above) from the study.

  1. Create a vector of the id data called id using the c() function.

  2. Now, create the id vector again, but this time use the a:b function.

  3. Now create the id vector again! But this time use the seq() function. To get help on this function, look at the help menu with ?seq

  4. Create a vector of the before drink data called before using c().

  5. Create a vector of the after drink data called after using c().

  6. Create a vector of the age data called age using c().

  7. Create a vector of the sex data called sex but don’t use just the c() function (that would be a lot of typing…). Instead, just repeat the vector c("male", "female") several times using the rep() function.

  8. Create a vector of the eye color data called eye_color using the rep() function.

Combining and changing vectors

  1. Create a new vector called age_months that shows the participants’ age in months instead of years. (Hint: Just multiply each age value by 12)

  2. Oops! It turns out that the watch used to measure time was off. All the before times are 1 second too fast, and all the after times are 1 second too slow. Correct these values by using simple arithmetic and then (re)assigning the objects with <-!

  3. Create a new vector called change that shows the change in participants’ scores from before to after (Hint: Just subtract one vector from the other)

  4. Create a new vector called average that shows the participants’ average score across both tests. That is, the first element of average should be the average of the first participant’s two scores, and the second element should be the average of the second participant’s two scores…(Hint: Don’t use the mean() function! Instead, use basic arithmetic with + and /. That is, the elements of average should be before plus after divided by 2.)

Applying functions to vectors

  1. How many elements are in each of the original data vectors? (Hint: use length()). If the number of elements in each is not the same, you typed something in wrong!

  2. What was the standard deviation of ages? Assign the result to a scaler object called age_sd.

  3. What is the median age? Assign the result to a scaler object called age_median.

  4. How many people were there of each sex? (Hint: use table())

  5. What percent of people had each sex? (Hint: use table() then divide by its sum with sum())

  6. Calculate the mean of the sex column. What happens and why?

  7. What was the mean before time? Assign the result to a scaler object called before_mean.

  8. What was the mean after time? Assign the result to a scaler object called after_mean.

  9. What was the difference in the mean before times and the mean after times? Calculate this in two ways: once using the change vector, and once using the before_mean and after_mean objects. You should get the same answer for both!

Checkpoint!!!

Standardizing (z-scores) vectors

  1. Create a vector called before_z showing a standardized version of before. (Hint: Standardizing a variable means subtracting the mean, and then dividing the result by the standard deviation.).

  2. Create a vector called after_z showing a standardized version of after.

  3. What was the largest before score? What was the largest before_z score?

  4. What was the smallest after score? What was the smallest after_z score?

  5. What should the mean and standard deviation of before_z and after_z be? Test your predictions by making the appropriate calculations.

Random samples from distributions

R has lots of functions for drawing random samples from probability distributions. For example, you can draw random samples from a vector with sample(), or draw random samples of values from a Normal distribution using the rnorm(n, mean, sd) function. Here are some examples:

# Draw 10 random numbers from the integers 1 to 100
sample(x = 1:100, size = 10)

# Simulate 10 flips from a fair coin
sample(x = c("H", "T"), size = 10, replace = TRUE)

# Random sample of 50 values from a Normal distribution with mean = 20 and sd = 10
rnorm(n = 50, mean = 20, sd = 10)

  1. Create a vector called samp_10 that contains 10 samples from a Normal distribution with a mean of 100 and a standard deviation of 10.

  2. Create a vector called samp_1000 that contains 1,000 samples from the same Normal distribution as above (that is, also with a mean of 100 and standard deviation of 10).

  3. Before making any calculations, what would you guess the mean and standard deviations of samp_10 and samp_1000 are? If your predictions are the same, which vector’s mean and standard deviation do you expect to be closer to your predictions?

  4. Calculate the mean and standard deviations of samp_10 and samp_1000 separately. Was your prediction correct?

  5. Simulate 100 flips from a fair coin using sample() (Hint: include the arguments x = c("H", "T"), size = 100, replace = TRUE)

  6. Simulate 100 flips from a biased coin where the probability of heads is 0.8 and the probability of tails is 0.2 (Hint: You can do this in two ways, either by including more heads than tails in the x argument, or by using the prob argument. Look at the help menu for the sample function for help.)

Bonus: The Room with 100 Boxes

Here is a fun little risky decision making game you can program in R using the sample() function. Imagine the following. There is a room with 100 boxes. 99 of the 100 boxes each contain 10 Thousand EUR, while 1 of the boxes contains a bomb.

Here’s the question…if you walked into the room with 100 boxes, how many would you want to open? If you don’t get the bomb, you can keep all of the money in the boxes you open. If you get the bomb, you get nothing (and die).

The code below will create a plot of the boxes game. If you’d like to, you could try running it in your R session to see the result.

# Plot of the Boxes Game

# Plotting space
plot(1, 
     xlim = c(0, 11), 
     ylim = c(0, 11),
     xlab = "", ylab = "", main = "The 100 Boxes Game!", 
     type = "n", xaxt = "n", yaxt = "n")

text(x = 5.5,
     y = 11, 
     labels = "There are 100 boxes\n99 / 100 contain 10,000 EUR (each) and 1 / 100 contains a bomb! How many will you open?", 
     font = 3,    # Italic font
     cex = .8)    # Slightly smaller font size

# Boxes
points(x = rep(1:10, times = 10), 
       y = rep(1:10, each = 10), 
       pch = 22, 
       cex = 4, 
       bg = sample(c(rep("green", 99), "red")))

# Labels
points(x = rep(1:10, times = 10), 
       y = rep(1:10, each = 10), 
       pch = "?")

Here’s how you can play the boxes game in R. First, create the room as an object room_100 which contains a vector with 99 values of 10 (representing 10 Thousand Euros) and one value of negative infinity (-Inf) which represents the bomb.

# This vector represents the room of 100 boxes
room_100 <- c(rep(10, 99), -Inf)

First, put the number of boxes you want to open as a new scaler object called open:

open <- 0  # How many do you want to open?

Now run the following code to see what you get!

# Play the Room with 100 Boxes Game!

result <- sample(x = room_100,  # Sample from the room...
                 size = open)

# Print what you got!

result       # Show what's in each box (1 means 10,000 EUR)
sum(result)  # Your total winnings!

You can also represent the boxes game by writing your own custom function in R. Run the following chunk to create the new function boxes_game. The code uses advanced functions like if() and function() that we haven’t learned yet, but feel free to take a closer look to try to understand the logic.

When you run the following code, ‘nothing’ will happen. But in fact, you are defining a new function called boxes_game that you can use later to actually play the game.

# Run this chunk to create the function
boxes_game <- function(open, 
                       room) {
  
# Outcome if no boxes are opened
  
if(open == 0) {
  
  print("You didn't open any boxes! You earned nothing but are still alive")}

# If at least 1 box is opened...
  
if(open > 0) {
  
  # Calculate the result
  result <- sample(x = room,
                   size = open)

# If -Inf (the bomb) is in the result...
if(-Inf %in% result) {
  
  print(paste("You're dead!!! You opened ", open, 
              " boxes and got the bomb!!!", sep = ""))}

# If -Inf (the bomb) is NOT in the result...
if((-Inf %in% result) == FALSE) {
  
  print(paste("Congratulations!!! You opened ", open, 
              " boxes and earned ", sum(result), 
              " Thousand Euros! Don't you want to play again? :)", sep = ""))}
}
  
}

Now you have defined the new function boxes_game(). To play the game, evaluate the function by specifying the two arguments: open is the the number of boxes you want to open, and room defines the room! For example, here’s how you’d play the game by opening 5 boxes in the room with 100 boxes:

# Play boxes game with 5 boxes in room_100

boxes_game(open = 5, 
           room = room_100)

Play the game a few times and see how you do. When you are done, try creating another room called room_risky that contains only 10 values: 9 values of 1000 Thousand (aka, 1 Million) and 1 bomb. Try playing the game in this room a few times and see how your results change.

Submit!