Lab Name

library(tidyverse)
library(openintro)
library(infer)

Getting started

us_adults <- tibble(climate_change_affects = c(rep("Yes", 62000), rep("No", 38000)))
ggplot(us_adults, aes(x = climate_change_affects)) +
  geom_bar() +
  labs(
    x = "", y = "",
    title = "Do you think climate change is affecting your local community?"
  ) +
  coord_flip()

us_adults %>%
  count(climate_change_affects) %>%
   mutate(p = n/sum(n))

## # A tibble: 2 × 3
##   climate_change_affects     n     p
##   <chr>                  <int> <dbl>
## 1 No                     38000  0.38
## 2 Yes                    62000  0.62

n <- 60
samp <- us_adults %>%
  sample_n(size = n)

Confidence levels

…

samp %>%
  specify(response = climate_change_affects, success = "Yes") %>%
  generate(reps = 1000, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.95)

## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1    0.533    0.767

# Question responses:
# Stating that the confidence interval is 95% means that there is a 95% chance that the true proportion is contained within the interval.
# Given that my confidence interval ranges in between 0.433 and 0.683, the true population proportion of 0.62 is contained within the interval.
# I would expect 95% of the intervals to capture the true population proportion, as it is a 95% confidence interval. This makes it reasonable to assume that given a large enough amount of opportunities, 95% of the intervals generated will contain the true proportion.
# C:/Users/emmap/AppData/Local/Temp/Untitled.png
# 47 out of 50 generated intervals successfully capture the true proportion, which is equivalent to around 94%. It is not exactly equal to the confidence level, but this is likely because 50 is a relatively small amount of intervals to generate. A larger amount such as 100 or 100 would likely have ended up being closer to the confidence level.

More practice

#Question responses
# Given a confidence interval of 0.96, I would expect it to be wider because a greater amount of its intervals would be accurately capturing the proportion. In order to increase the likelihood of this occurring, the average interval would have to contain more values.
samp %>%
  specify(response = climate_change_affects, success = "Yes") %>%
  generate(reps = 1000, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.99)

## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1    0.483    0.800

# I am 99% confident that between 38.3% and 73.3% of US adults believe that climate change is impacting their community.
# When I used the app again at the 99% confidence interval, 49 out of 50 of the intervals contained the true proportion. This is equivalent to 98%, which is slightly lower than the confidence level.
# I am going to utilize a confidence interval of 90%, which I expect to be narrower than both the 95% and 99% intervals I've previously experimented with.
samp %>%
  specify(response = climate_change_affects, success = "Yes") %>%
  generate(reps = 1000, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.90)

## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1     0.55     0.75

# When using the app for the 90% confidence interval, 45 out of 50 intervals generated contained the true proportion. This is equivalent to 90%, which does indeed match the confidence level.
# As the sample size increases, the intervals become wider and generally more accurate; an increase in sample size corresponds with a decrease in the amount of intervals that fail to capture the true proportion.
# As the amount of bootstrap samples increases, the width of the intervals also increases.