Sample proportion value effects on bootstrap CIs

One additional element that changes the width of the confidence interval is the sample parameter value, \(\hat{p}\).

Generally, when the true parameter is close to 0.5, the standard error of \(\hat{p}\) is larger than when the true parameter is closer to 0 or 1. When calculating a bootstrap t-confidence interval, the standard error controls the width of the CI, and here (given a true parameter of 0.8) the sample proportion is higher than in previous exercises, so the width of the confidence interval will be narrower.

This exercise is part of the course

Foundations of Inference in R

View Course

Exercise instructions

calc_p_hat() is shown in the script to calculate the sample proportions. calc_t_conf_int() from the previous exercise has been updated to now use any value of p_hat as an argument. Read their definitions and try to understand them.
Run the code to calculate the bootstrap t-confidence interval for the original population.
Consider a new population where the true parameter is 0.8, one_poll_0.8. Calculate \(\hat{p}\) of this new sample, using the same technique as with the original dataset. Call it p_hat_0.8.
Find the bootstrap t-confidence interval using the new bootstrapped data, one_poll_boot_0.8, and the new \(\hat{p}\). Notice that it is narrower than previously calculated.

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

calc_p_hat <- function(dataset) {
  dataset %>%
    summarize(stat = mean(vote == "yes")) %>%
    pull()
}
calc_t_conf_int <- function(resampled_dataset, p_hat) {
  resampled_dataset %>%
    summarize(
      lower = p_hat - 2 * sd(stat),
      upper = p_hat + 2 * sd(stat)
    )
}

# Find proportion of yes votes from original population
p_hat <- calc_p_hat(one_poll)

# Review the value
p_hat  

# Calculate bootstrap t-confidence interval (original 0.6 param)
calc_t_conf_int(one_poll_boot, p_hat)

# Find proportion of yes votes from new population
p_hat_0.8 <- ___
  
# Review the value
p_hat_0.8  
  
# Calculate the bootstrap t-confidence interval (new 0.8 param)
___

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