If you have access to data on an entire population, say the opinion of every adult in the United States on whether or not they think climate change is affecting their local community, it’s straightforward to answer questions like, “What percent of US adults think climate change is affecting their local community?”. Similarly, if you had demographic information on the population you could examine how, if at all, this opinion varies among young and old adults and adults with different leanings. If you have access to only a sample of the population, as is often the case, the task becomes more complicated. What is your best guess for this proportion if you only have data from a small sample of adults? This type of situation requires that you use your sample to make inference on what your population looks like.

Setting a seed: You will take random samples and build sampling distributions in this lab, which means you should set a seed on top of your lab. If this concept is new to you, review the lab on probability.

Getting Started

Load packages

In this lab, we will explore and visualize the data using the tidyverse suite of packages, and perform statistical inference using infer.

Let’s load the packages.

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

The data

A 2019 Pew Research report states the following:

To keep our computation simple, we will assume a total population size of 100,000 (even though that’s smaller than the population size of all US adults).

Roughly six-in-ten U.S. adults (62%) say climate change is currently affecting their local community either a great deal or some, according to a new Pew Research Center survey.

Source: Most Americans say climate change impacts their community, but effects vary by region

In this lab, you will assume this 62% is a true population proportion and learn about how sample proportions can vary from sample to sample by taking smaller samples from the population. We will first create our population assuming a population size of 100,000. This means 62,000 (62%) of the adult population think climate change impacts their community, and the remaining 38,000 does not think so.

set.seed(1234)
us_adults <- tibble(
  climate_change_affects = c(rep("Yes", 62000), rep("No", 38000))
)

The name of the data frame is us_adults and the name of the variable that contains responses to the question “Do you think climate change is affecting your local community?” is climate_change_affects.

We can quickly visualize the distribution of these responses using a bar plot.

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()

We can also obtain summary statistics to confirm we constructed the data frame correctly.

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

In this lab, you’ll start with a simple random sample of size 60 from the population.

n <- 60
samp <- us_adults %>%
  sample_n(size = n)
  1. What percent of the adults in your sample think climate change affects their local community? Hint: Just like we did with the population, we can calculate the proportion of those in this sample who think climate change affects their local community.
samp %>%
  count(climate_change_affects) %>%
  mutate(p = n /sum(n))
## # A tibble: 2 × 3
##   climate_change_affects     n     p
##   <chr>                  <int> <dbl>
## 1 No                        23 0.383
## 2 Yes                       37 0.617

In this sample, approximately 61.7% of U.S. adults think climate change affects their local community, which is close to the 62% who think climate change affects their local community.

  1. Would you expect another student’s sample proportion to be identical to yours? Would you expect it to be similar? Why or why not?

I think another student will have an identical value because the set.seed() function ensures that the random values are reproduced each time the code chunk is run. The value will be close to the population proportion of people who think climate change affects their community.

Confidence intervals

Return for a moment to the question that first motivated this lab: based on this sample, what can you infer about the population? With just one sample, the best estimate of the proportion of US adults who think climate change affects their local community would be the sample proportion, usually denoted as \(\hat{p}\) (here we are calling it p_hat). That serves as a good point estimate, but it would be useful to also communicate how uncertain you are of that estimate. This uncertainty can be quantified using a confidence interval.

One way of calculating a confidence interval for a population proportion is based on the Central Limit Theorem, as \(\hat{p} \pm z^\star SE_{\hat{p}}\) is, or more precisely, as \[ \hat{p} \pm z^\star \sqrt{ \frac{\hat{p} (1-\hat{p})}{n} } \]

Another way is using simulation, or to be more specific, using bootstrapping. The term bootstrapping comes from the phrase “pulling oneself up by one’s bootstraps”, which is a metaphor for accomplishing an impossible task without any outside help. In this case the impossible task is estimating a population parameter (the unknown population proportion), and we’ll accomplish it using data from only the given sample. Note that this notion of saying something about a population parameter using only information from an observed sample is the crux of statistical inference, it is not limited to bootstrapping.

In essence, bootstrapping assumes that there are more of observations in the populations like the ones in the observed sample. So we “reconstruct” the population by resampling from our sample, with replacement. The bootstrapping scheme is as follows:

  • Step 1. Take a bootstrap sample - a random sample taken with replacement from the original sample, of the same size as the original sample.
  • Step 2. Calculate the bootstrap statistic - a statistic such as mean, median, proportion, slope, etc. computed on the bootstrap samples.
  • Step 3. Repeat steps (1) and (2) many times to create a bootstrap distribution - a distribution of bootstrap statistics.
  • Step 4. Calculate the bounds of the XX% confidence interval as the middle XX% j knof the bootstrap distribution.

Instead of coding up each of these steps, we will construct confidence intervals using the infer package.

Below is an overview of the functions we will use to construct this confidence interval:

Function Purpose
specify Identify your variable of interest
generate The number of samples you want to generate
calculate The sample statistic you want to do inference with, or you can also think of this as the population parameter you want to do inference for
get_ci Find the confidence interval

This code will find the 95 percent confidence interval for proportion of US adults who think climate change affects their local community.

#set.seed(1234)
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    0.483     0.75
  • In specify we specify the response variable and the level of that variable we are calling a success.
  • In generate we provide the number of resamples we want from the population in the reps argument (this should be a reasonably large number) as well as the type of resampling we want to do, which is "bootstrap" in the case of constructing a confidence interval.
  • Then, we calculate the sample statistic of interest for each of these resamples, which is proportion.

Feel free to test out the rest of the arguments for these functions, since these commands will be used together to calculate confidence intervals and solve inference problems for the rest of the semester. But we will also walk you through more examples in future chapters.

To recap: even though we don’t know what the full population looks like, we’re 95% confident that the true proportion of US adults who think climate change affects their local community is between the two bounds reported as result of this pipeline.

Confidence levels

  1. In the interpretation above, we used the phrase “95% confident”. What does “95% confidence” mean?

In this case, you have the rare luxury of knowing the true population proportion (62%) since you have data on the entire population.

The phrase “95% confidence” means that there is 95% confidence that the true population proportion of a given sample is between a specific range. In this scenario, we are 95% confident that the true population proportion of the samples will be between 50% and 75%, with the range changing slightly with each calculation.

  1. Does your confidence interval capture the true population proportion of US adults who think climate change affects their local community? If you are working on this lab in a classroom, does your neighbor’s interval capture this value?

This confidence interval captures the true population proportion of U.S. adults who believe climate change affects their local community, which is 62%. The confidence interval range may be slightly different if the set.seed() function isn’t used. Nevertheless, the value of the true population proportion would be captured by the interval.

  1. Each student should have gotten a slightly different confidence interval. What proportion of those intervals would you expect to capture the true population mean? Why?

I would expect every interval to capture the true population mean, since each student will use the same number of reps and type of samples to compute the confidence interval.

In the next part of the lab, you will collect many samples to learn more about how sample proportions and confidence intervals constructed based on those samples vary from one sample to another.

  • Obtain a random sample.
  • Calculate the sample proportion, and use these to calculate and store the lower and upper bounds of the confidence intervals.
  • Repeat these steps 50 times.

Doing this would require learning programming concepts like iteration so that you can automate repeating running the code you’ve developed so far many times to obtain many (50) confidence intervals. In order to keep the programming simpler, we are providing the interactive app below that basically does this for you and created a plot similar to Figure 5.6 on OpenIntro Statistics, 4th Edition (page 182).

  1. Given a sample size of 60, 1000 bootstrap samples for each interval, and 50 confidence intervals constructed (the default values for the above app), what proportion of your confidence intervals include the true population proportion? Is this proportion exactly equal to the confidence level? If not, explain why. Make sure to include your plot in your answer.

Of the 50 confidence intervals constructed, 47 of the confidence intervals include the true population proportion, which is 94%. This is not exactly equal to the 95% confidence level that was set for the number of samples and sample size amounts.

More Practice

  1. Choose a different confidence level than 95%. Would you expect a confidence interval at this level to be wider or narrower than the confidence interval you calculated at the 95% confidence level? Explain your reasoning.

I would expect that the confidence interval at 85% would be more narrow that the confidence level calculated at 95% because the confidence level is smaller and therefore more difficult to capture the true population proportion.

  1. Using code from the infer package and data format the one sample you have (samp), find a confidence interval for the proportion of US Adults who think climate change is affecting their local community with a confidence level of your choosing (other than 95%) and interpret it.
samp %>%
  specify(response = climate_change_affects, success = "Yes") %>%
  generate(reps = 1000, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.85)
## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1    0.533      0.7

Using an 85% confidence level, the confidence interval range is from 51.7% to 70%, meaning that we can say with an 85% confidence level that the true population proportion of people who believe climate change affects their local community will be captured between these two intervals.

  1. Using the app, calculate 50 confidence intervals at the confidence level you chose in the previous question, and plot all intervals on one plot, and calculate the proportion of intervals that include the true population proportion. How does this percentage compare to the confidence level selected for the intervals?

Of the 50 confidence intervals at an 85% confident level, 41 of the intervals capture the true population proportion, which is 82%. This is 3% lower than the 85% confidence level that was set.

  1. Lastly, try one more (different) confidence level. First, state how you expect the width of this interval to compare to previous ones you calculated. Then, calculate the bounds of the interval using the infer package and data from samp and interpret it. Finally, use the app to generate many intervals and calculate the proportion of intervals that are capture the true population proportion.
samp %>%
  specify(response = climate_change_affects, success = "Yes") %>%
  generate(reps = 1000, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.75)
## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1     0.55    0.683

Since this confidence level is smaller than the previous confidence level, I expect a more narrow confidence interval.

When setting the level of get_ci to 0.75, the lower and upper confidence intervals are 0.55 and 0.683, respectively, which is narrower than the previous interval, which was 0.517 and 0.70. When using the app to generate 5 different intervals with a confidence level of 75%, the percentages of confidence intervals that captured the true population proportion were, respectively, 76%, 68%, 82%, 52%, and 60%.

  1. Using the app, experiment with different sample sizes and comment on how the widths of intervals change as sample size changes (increases and decreases).

Continuing with a confidence interval of 75%, there wasn’t a discernible pattern in terms of the amount of confidence intervals that captured the true population proportion with varying sample sizes. Decreasing the the sample size from 60 to 50 resulted in a 78% confidence interval, meaning that 78% of the confidence intervals captured the true population proportion. When increasing the sample size to 100 and running the app twice, the confidence interval decreased slightly to 72% and 68%, respectively. When increasing the sample size to 500 and running the app twice, the confidence interval was 78% and 70% respectively. Finally, when running the app twice with a sample size of 30, the confidence intervals were 64% and 76%, respectively. The smaller the sample size, the wider the intervals, while the intervals become narrow when the sample size increases.

  1. Finally, given a sample size (say, 60), how does the width of the interval change as you increase the number of bootstrap samples. Hint: Does changing the number of bootstrap samples affect the standard error?
s_e <- function(x) sd(x/sqrt(length(x)))
#bootstrap set to 500
samp_1 <- samp %>%
  specify(response = climate_change_affects, success = "Yes") %>%
  generate(reps = 500, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.75)
s_e(samp_1)
## [1] 0.071875
#bootstrap set to 1500
samp_2 <- samp %>%
  specify(response = climate_change_affects, success = "Yes") %>%
  generate(reps = 1500, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.75)
s_e(samp_2)
## [1] 0.06666667
#bootstrap set to 2500
samp_3 <- samp %>%
  specify(response = climate_change_affects, success = "Yes") %>%
  generate(reps = 2500, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.75)
s_e(samp_3)
## [1] 0.06666667

When changing the number of bootstrap samples, there doesn’t appear to be a significant change in the width of the intervals. Changing the number of bootstrap samples does not have an affect on the standard error. However, the standard error would be affected by a change in sample size.