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.

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.

set.seed(34)
n <- 60
samp <- us_adults %>%
  sample_n(size = n)

ggplot(samp, aes(x = climate_change_affects)) +
  geom_bar() +
  labs(
    x = "", y = "",
    title = "Do you think climate change is affecting your local community?"
  ) +
  coord_flip() 

samp %>%
  count(climate_change_affects) %>%
  mutate(p = n /sum(n))
## # A tibble: 2 × 3
##   climate_change_affects     n     p
##   <chr>                  <int> <dbl>
## 1 No                        24   0.4
## 2 Yes                       36   0.6
  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. #ANSWER #1 60% believes it affects their lives. Code is below.
set.seed(34)
n <- 60
samp <- us_adults %>%
  sample_n(size = n)

ggplot(samp, aes(x = climate_change_affects)) +
  geom_bar() +
  labs(
    x = "", y = "",
    title = "Do you think climate change is affecting your local community?"
  ) +
  coord_flip() 

samp %>%
  count(climate_change_affects) %>%
  mutate(p = n /sum(n))
## # A tibble: 2 × 3
##   climate_change_affects     n     p
##   <chr>                  <int> <dbl>
## 1 No                        24   0.4
## 2 Yes                       36   0.6
  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?

#ANSWER #2 It is highly unlikely it will be identical to ours however it will most likely be extremely similar as the sample size is large enough to be somewhat representative of the entire population so it will be close to the population proportion and our results, with some margin of error of course.

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:

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.

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    0.467    0.717

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. #3 Answer #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 which is 46% and 71% of the population. There is however a 5% chance that the true proportion of US adults who think climate change affects their local community is LESS than 46% and MORE than 71%. this confidence interval gives us a range of outcomes ( 46 -71 % ) of which we are relatively certain that the true proportion lies within the bounds of the range.

  2. 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? #4 Answer : Yes it does. Our interval is 46-71%, and the true population proportion is 62% which is within the bounds of the range.

  3. 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?

#5 Answer : I would expect the vast majority of them to capture the true population mean given that we are using 1000 repetitions with a sample size of 60. To be exact, I expect 95% of them to capture the true population mean because that is exactly what a 95% confidence interval means, you are going to capture the true population mean 95% of the time. So if you have 1,000 students, you would expect 950 of them to capture the mean with 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.

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.

#Answer 6 I am not sure how to include the plot, but only 10% of the confidences intervals DO NOT include the true population proportion, 90% do. This is not equal to the confidence level. I am not sure why to be honest. Is it possible that there is some “margin of error” among the confidence intervals, so that the 95% confidence interval has a margin of error of +/- 5?%

More Practice

  1. Choose a different confidence level than 95%. Would you expect a confidence interval at this level to me wider or narrower than the confidence interval you calculated at the 95% confidence level? Explain your reasoning. #Answer 7 : I chose 99% confidence interval. I would expect the confidence interval to be wider because we are asking for a higher degree of certainty without increasing the number of samples taken, so we would need to widen the confidence interval to be 99% sure that the true mean of the population is included in the confidence interval ( with 99% confidence of course)

  2. Using code from the infer package and data fromt 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.

#Answer 8 - we use a 99% confidence interval. What we are saying is that, by taking 1,000 bootstrap samples, with sample size of 60, we can be 99% sure that the true population proportion of US Adults who think climate change is affecting their local community is somewhere between 41.65% and 75%

set.seed(34)
n <- 60
samp <- us_adults %>%
  sample_n(size = n)

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    0.417     0.75
  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?

#Answer 9. 100% of the intervals include the true population mean.

  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. #Answer 10 - we are using a 90% confidence interval so we would expect the confidence intervals to be narrower because we are get a less precise estimate which is more prone to error. We can see it clearly, when comparing the output of this R code to the previous question #8.
set.seed(34)
n <- 60
samp <- us_adults %>%
  sample_n(size = n)

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    0.483      0.7
  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).

#Answer 11, as sample size increases, width of confidence intervals decreases and vice versa

  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 bootstap samples affect the standard error? #Answer 12, as bootstramp sample size increases, width of confidence intervals decreases and vice versa