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.
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)To create your new lab report, in RStudio, go to New File -> R Markdown… Then, choose From Template and then choose Lab Report for OpenIntro Statistics Labs from the list of templates.
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 x 3
## climate_change_affects n p
## <chr> <int> <dbl>
## 1 No 38000 0.38
## 2 Yes 62000 0.62In 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)#set.seed(1117)
samp %>%
count(climate_change_affects) %>%
mutate(p = n /sum(n))the proportion of people who who think climate change affects their local community is 49% of the sample. 2. 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 would not expect another student’s proportion to be identical, but possibly similar. The reason for that is that our sample are random so it cannot be identical.
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)specify we specify the response variable and the level of that variable we are calling a success.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.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.
In the interpretation above, we used the phrase “95% confident”. What does “95% confidence” mean? A 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population. In this case, you have the rare luxury of knowing the true population proportion (62%) since you have data on the entire population.
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? Yes, it would be similar value.
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 to capture the true population mean because of the 95% confidence interval certainty.
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).
I would expect the interval to be narrower because the confidence level is lower.
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 = 0.75)I chose 75% confidence level and the boundaries were very narrow. The distribution became very narrow because of lower confidence level, that the sample is not be truth.
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? It is still narrow, as less than 95% confidence level the distribution becomes narrower, but it looks very similar to the sample size 50 or much higher.
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 = 0.35)I am expecting to be very narrow, as confidence level is at 35%. It is visible the uncertainty of data at such a low confidence level.
Using the app, experiment with different sample sizes and comment on how the widths of intervals change as sample size changes (increases and decreases). With different sample size as 40 an 400 I experience very similar output with the same confidence level,
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? In my example of sample size 60 the higher bootrap sample the broader and better distribution of the true population proportion. * * *