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)
library(ggplot2)

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.

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.

Insert your answer here

samp %>%
  count(climate_change_affects) %>%
  mutate(p = n /sum(n))
## # A tibble: 2 × 3
##   climate_change_affects     n     p
##   <chr>                  <int> <dbl>
## 1 No                        21  0.35
## 2 Yes                       39  0.65
if the sample is 60, I’d assume 62% of those 60 would say climate change affects them. In my random sample, 45/60 said it affects them, so 75%
  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?

Insert your answer here

I’d expect another student’s sample proportion to be rather similar, but not exactly the same. The random seed would affect the exact amounts of our proportions.

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.

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    0.517    0.767
  • 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?

Insert your answer here

It means that from that confidence interval calculation, that 95% of the intervals sampled contain the actual proportion of the full sample of US adults who believe in climate change affecting their local community, within their lower and upper interval values.

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

  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?

Insert your answer here

Yes it does. The true proportion is 62%, and in my samp example, it was 65%. The confidence intervals I generated were 53.3% and 76.7%, which does contain the true proportion.
  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?

Insert your answer here

I’d expect 95% of them to do so. While all the intervals would be slightly different, 95% of them would contain the true population proportion of 0.62 to be within their lower_ci and upper_ci ranges.

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.

Insert your answer here

# Sample
n <- 60
samp2 <- us_adults %>%
  sample_n(size = n)

samp2 %>%
  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     0.55    0.783
# Ranges from lower_ci to upper_ci include the true proportion 95% of time

ggplot(samp2, aes(x = climate_change_affects)) +
  geom_bar() +
  labs(
    x = "", y = "",
    title = "Do you think climate change is affecting your local community? (Sample 2)"
  ) +
  coord_flip() +
  geom_text(aes(label = after_stat(count)), stat = "count", hjust = 3, colour = "black")

Given the above parameters in the Shinyapp, only 2 samples of 50 (4%) lack the true poportion (0.62) so 96% of the intervals contain the true poportion within their lower and upper confidence intervals. This is nigh identical to the the 95% confidence level we set.
* * *
## More Practice
7. 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.
Insert your answer here
r samp2 %>% specify(response = climate_change_affects, success = "Yes") %>% generate(reps = 50, type = "bootstrap") %>% calculate(stat = "prop") %>% get_ci(level = 0.95)
## # A tibble: 1 × 2 ## lower_ci upper_ci ## <dbl> <dbl> ## 1 0.587 0.833
```r # 0.554 to 0.783
samp2 %>% specify(response = climate_change_affects, success = “Yes”) %>% generate(reps = 50, type = “bootstrap”) %>% calculate(stat = “prop”) %>% get_ci(level = 0.5) ```
## # A tibble: 1 × 2 ## lower_ci upper_ci ## <dbl> <dbl> ## 1 0.633 0.717
```r # 0.633 to 0.712
samp2 %>% specify(response = climate_change_affects, success = “Yes”) %>% generate(reps = 50, type = “bootstrap”) %>% calculate(stat = “prop”) %>% get_ci(level = 0.1) ```
## # A tibble: 1 × 2 ## lower_ci upper_ci ## <dbl> <dbl> ## 1 0.65 0.667
r # 0.633 to 0.712
###### If I choose a lower confidence interval, then I would expect the range from the lower to upper ci’s to be narrower. As the range decreases, so does the likelihood of the true proportion to lie within the confines of those ranges. The lower our confidence interval, the closer we get to the true p_hat. Remember that we’re pulling 50 different intervals here, so lowering our confidence level means more of them will not contain p_hat, which we can see with the interactive shiny app above.
  1. Using code from the infer package and data from 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.

Insert your answer here

samp %>%
  specify(response = climate_change_affects, success = "Yes") %>%
  generate(reps = 1000, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.33)
## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1    0.617    0.683
samp %>%
  count(climate_change_affects) %>%
  mutate(p = n /sum(n))
## # A tibble: 2 × 3
##   climate_change_affects     n     p
##   <chr>                  <int> <dbl>
## 1 No                        21  0.35
## 2 Yes                       39  0.65
Using a confidence level of 33%, this gives me the lower_ci and upper_ci of 0.567 and 0.633 respectively. What this means, is that only 33% of the intervals will actually contain the true proportion of 0.62. The remaining 67% will either have ranges that are too high or low and do not contain p_hat.
  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?

Insert your answer here

# For a 95% ci, qnorm(0.95 / 2) = qnorm((1 - 0.95)/2) = qnorm(0.025) = -1.959964  because of symmetry.
# Remember we need to consider the two tails, 2.5% to the left, 2.5% to the right, Divide by 2 for left and right tail.
# For a 95% confidence interval, critical value = 1.96 (https://fall2023.data606.net/slides/05-Foundation_for_Inference.pdf)

true_prop <- 0.62
num_intervals <- 50
# Let's use ci = 0.33
my_ci <- 0.33
crit_value <- abs(qnorm((1 - my_ci)/2))
# Gives us critical value of 0.426148

# Create an empty frame for our plot
ci <- data.frame(mean = numeric(num_intervals), min = numeric(num_intervals), max = numeric(num_intervals))  

for (i in seq_len(num_intervals)) {
  samp_i <- sample(us_adults$climate_change_affects == "Yes", size = num_intervals, replace = TRUE)
  se <- sd(samp_i) / sqrt(length(samp_i)) # standard error
  # se = sqrt(p_hat * (1 - p_hat)/ n)    is more accurate where p_hat = n /sum(n)  but are basically the same
  ci[i,] <- c(mean(samp_i),  # Adding elements to the empty frame
              mean(samp_i) - crit_value * se,
              mean(samp_i) + crit_value * se)
}

ci$sample <- 1:nrow(ci)
ci$sig <- ci$min < true_prop & ci$max > true_prop #Adding the True False to see if the ci's contain the true proportion 0.62


# Plot all the confidence intervals using the dataframe we just made
ggplot(ci, aes(x=min, xend= max, y=sample, yend=sample, color= sig)) +
  geom_vline(xintercept = true_prop) +
  geom_segment() + xlab('CI') + ylab('') +
  scale_color_manual(values = c('TRUE' = 'blue', 'FALSE' = 'red'))

  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 capture the true population proportion.

Insert your answer here

true_prop <- 0.62
num_intervals <- 50
# Let's use ci = 0.85
my_ci2 <- 0.85
crit_value2 <- abs(qnorm((1 - my_ci2)/2))


# Create an empty frame for our plot
ci2 <- data.frame(mean = numeric(num_intervals), min = numeric(num_intervals), max = numeric(num_intervals))  

for (i in seq_len(num_intervals)) {
  samp_i2 <- sample(us_adults$climate_change_affects == "Yes", size = num_intervals, replace = TRUE)
  se2 <- sd(samp_i2) / sqrt(length(samp_i2)) # standard error
  # se = sqrt(p_hat * (1 - p_hat)/ n)    is more accurate where p_hat = n /sum(n)  but are basically the same
  ci2[i,] <- c(mean(samp_i2),  # Adding elements to the empty frame
              mean(samp_i2) - crit_value2 * se2,
              mean(samp_i2) + crit_value2 * se2)
}

ci2$sample <- 1:nrow(ci2)
ci2$sig <- ci2$min < true_prop & ci2$max > true_prop #Adding the True False to see if the ci's contain the true proportion 0.62


# Plot all the confidence intervals using the dataframe we just made
ggplot(ci2, aes(x=min, xend= max, y=sample, yend=sample, color= sig)) +
  geom_vline(xintercept = true_prop) +
  geom_segment() + xlab('CI') + ylab('') +
  scale_color_manual(values = c('TRUE' = 'blue', 'FALSE' = 'red'))

As sample size increases, the width of confidence intervals decreases and vice versa. The bounds of the interval for this sample will contain the true proportion, with 85% confidence for each interval. Since I chose a much larger confidence interval, I was correct in expecting the widths of the intervals to be larger than for my prior example of 0.33 . In this instance, looking at ci2 I can directly see how many confidence intervals contained the true proportion 0.62, which does change everytime I run this code as the sample is different each time, but it is always close to my chosen confidence interval of 0.85
11. Using the app, experiment with different sample sizes and comment on how the widths of intervals change as sample size changes (increases and decreases).
Insert your answer here
###### I started with a sample size of 50, at 1000 resamples and 95% confidence level. As the sample size increases, the width of the interval shrinks, meaning we’re having a smaller confidence interval cenetered around the true proportion . As the sample size decreases, the width of our interval is wider because the upper and lower values are more uncertain.
  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?

Insert your answer here

As the number of bootstrap resamples increases, the width of the intervals get wider, which makes sense given we’re sampling more data. However, the standard error will increase given that we’re casting a wider net with wider intervals. To minimize error, we want to be as close as we can to the true proportion (p_hat), which occurs when we choose a smaller confidence interval.