library(tidyverse)
library(tidymodels)
yawn <- read_csv("data/yawn.csv")

Bulletin

Learning goals

Use and understand simulation-based methods to …

Test for independence

  • First let’s, watch the experiment from Mythbusters.

  • Let \(t\) be the treatment group who saw a person yawn, \(c\) be the control group who did not see anyone yawn, and \(p\) be the proportion of people who yawned.

Exercise 11

We want to use simulation-based inference to assess whether or not yawning and seeing someone yawn are independent.

  • State the null and alternative hypotheses in words:

  • Select the appropriate null and alternative hypotheses written in mathematical notation:

  1. \(H_0: p_t = p_c \text{ vs. }H_a: p_t < p_c\)
  2. \(H_0: p_t = p_c \text{ vs. }H_a: p_t > p_c\)
  3. \(H_0: p_t = p_c \text{ vs. }H_a: p_t \neq p_c\)
  4. \(H_0: \hat{p}_t = \hat{p}_c \text{ vs. }H_a: \hat{p}_t < \hat{p}_c\)
  5. \(H_0: \hat{p}_t = \hat{p}_c \text{ vs. }H_a: \hat{p}_t > \hat{p}_c\)
  6. \(H_0: \hat{p}_t = \hat{p}_c \text{ vs. }H_a: \hat{p}_t \neq \hat{p}_c\)

Exercise 1.5

Type I and Type II error

Truth Reject \(H_0\) Fail to reject \(H_0\)
\(H_0\) is true Type 1 error (false +) Good decision
\(H_0\) is false Good decision Type 2 error (false -)

What does type I error look like in this case?

What does type II error look like?

Is type I or type II error more worrisome here?

How might we mitigate the more dangerous error?

Click here for further reading.

Exercise 2

Before using R to construct the null distribution, let’s generate the null distribution using playing cards! See AE-17 for the simulation instructions. You can also find them in the README of this application exercise.

Exercise 3

Uncomment the code to see read in the data from the class and visualize the null distribution.

#sim_data <- read_csv("https://sta199-f21-001.netlify.app/appex/data/yawn-sim.csv")
#ggplot(data = sim_data, mapping = aes(x = diff_in_prop)) +
#  geom_histogram(binwidth = 0.05) + 
#  labs(title = "Your Results: Difference in Proportion of Yawners")

  • What is the approximate center of the distribution? Is this what you expected? Why or why not?

  • The observed difference in proportions from the Mythbusters episode is 0.0441. Based on your simulated distribution, do yawning and seeing someone yawn appear to be dependent?

Exercise 4

Let’s use the data from the Mythbusters episode and simulation-based inference in R to test this claim. Based on their experiment, do yawning and seeing someone yawn appear to be dependent?

Evaluate this question using a simulation based approach. We will use the same null and alternative hypotheses as before. The data from Mythbusters is available in the yawn data frame.

  • Fill in the code below to generate the null distribution. Uncomment the code once it is complete.
set.seed(101921)
#null_dist <- yawn %>%
#  specify(response = ____, explanatory = _____, success = "yawn") %>%
#  hypothesize(null = "______") %>%
#  generate(100, type = "permute") %>%
#  calculate(stat = "______", 
#            order = c("trmt", "ctrl"))
  • Visualize the null distribution and shade in the area used to calculate the p-value.
# add code
  • Calculate p-value. Then use the p-value to make your conclusion using a significance level of 0.1.
# add code

Exercise 5

Do you believe the conclusions from this experiment? Why or why not?

  1. Simulation activity from Data science in a box↩︎