Confidence Intervals

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

set.seed(74229)

Exercise 1

55% of my sample or 33 people think climate change affects their local community.

us_adults <- tibble(
  climate_change_affects = c(rep("Yes", 62000), rep("No", 38000))
)

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

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

## # A tibble: 2 x 3
##   climate_change_affects     n     p
##   <chr>                  <int> <dbl>
## 1 No                        18   0.3
## 2 Yes                       42   0.7

Exercise 2

My sample statistics will be different than the sample statistics pulled by another student due to normal variability of samples - random data will vary from sample to sample. My sample proportion will be different if I ran sample simulation again. (hence setting seed) It isn’t really possible to say how different any one sample proportion could be from another sample proportion, only that with enough samples observed we could estimate a plausible range for population proportion.

Exercise 3

A 95% confidence level indicates the level of assurance we have that population parameter falls within this range. Said another way, 95% indicates the number of times out of 100 that the mean of the population will be within the given interval of the sample mean.

Exercise 4

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

## # A tibble: 1 x 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1      0.6    0.817

Yes, the confidence interval generated from performing 1000 repeated re-sampling simulations contains the actual population parameter of .62 in its CI bounds of .4167 and .667. With 1000 simulations, my neighbor’s caluclated CI, while possibly different than mine, will also capture this value per Central Limit Theorem.

Exercise 5

I would estimate that out of 100 intervals generated, 95 of those confidence intervals (or 95%) would contain the population parameter. This is how a 95% confidence level is interpreted.

Exercise 6

The shinyApp function would not run for me but if we set CI level we wanted in the bootstrapping simulation to 95%, and constructed 50 confidence intervals at this level, then roughly 47-48 of those confidence intervals generated should contain the population mean. This proportion may not be exactly equal to the confidence level since it is a simulation and the exact proportion of 95% of 50 is 47.5 which is not a possible value for number of CIs containing pop parameter.

Exercise 7

If we chose a confidence level that was higher than 95%, we would expect a wider confidence interval to capture data in tails. I.e. in order to be 99% confident that population mean is contained in CI, the plausable value range needs to increase.

If we chose a confidence level that was lower than 95%, our confidence interval would be narrower as we are getting a lower margin of error if we decrease the level of assurance we want to have of population metric being included in the confidence interval.

Exercise 8

samp %>%
  specify(response = climate_change_affects, success = "Yes") %>%
  generate(reps = 1000, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.90)

## # A tibble: 1 x 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1      0.6    0.783

The CI at 90% confidence level is narrower (.45, .65) than the CI at 95% (.416, .667) as we adjust the bounds to “shrink” with less need to account for a greater amount of confidence intervals to contain the population parameter. A 90% CI will account for 90 CIs generated containing population pop, vs a 95% CI that will need to estimate for 95 CIs generated containing population pop.

Exercise 9

If I change get_ci level arg to .99, I would see a wider confidence interval range with a lower lower_ci boundary and a larger upper_ci boundary. This would be to account for additional possible values in the tails of the distribution. A wider range like this means that while I can be 99% that population parameter will exist in the interval, I have lost some precision in my prediction of what the population parameter could be.

Exercise 10

The number of bootstrap samples will generally enable your confidence intervals to be more precise/narrower as standard error will decrease and the more sampling distribution sample mean data points you can generate, the better the approximation of distribution to normal distribution and the greater clustering around the true population mean will occur.

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