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)
library(DATA606)
##
## Welcome to CUNY DATA606 Statistics and Probability for Data Analytics
## This package is designed to support this course. The text book used
## is OpenIntro Statistics, 4th Edition. You can read this by typing
## vignette('os4') or visit www.OpenIntro.org.
##
## The getLabs() function will return a list of the labs available.
##
## The demo(package='DATA606') will list the demos that are available.
set.seed(28492)
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.
<- tibble(
us_adults 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.
<- 60
n <- us_adults %>%
samp sample_n(size = n)
Based on the results below, 71.7% of the sample thinks 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 17 0.283
## 2 Yes 43 0.717
I do not expect for the sample proportion to match another student’s sample because we each get a random selection of the population. I do however expect the sample proportions to be similar.
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 0.6 0.817
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 prop
ortion.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.
This 95% confidence interval is a range of values where we are 95% certain that it contains the true proportion 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.
Yes, the confidence interval captures the true population.
Each student would and should get a slightly different confidence interval because every sample from the population will be different. They should however capture the true population mean because at a 95% confidence level it is expected.
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).
Majority of the proportions of confidence intervals include the true population proportion. This proportion is close to equal to the 95% confidence level.
I chose a confidence level of 80%, sample size, bootstrap and confidence intervals are the same. Being that I changed the confidence level, I expect this level to be narrower than the 95% confidence interval. With a 15% decrease in the confidence level you can expect the values containing the true population proportion to also decrease.
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.**I am 80% confident that the true proportions of US Adults think climate change affects their local community is within the confidence intervals of (65%, 80%).
#80% confidence level
%>%
samp specify(response = climate_change_affects, success = "Yes") %>%
generate(reps = 1000, type = "bootstrap") %>%
calculate(stat = "prop") %>%
get_ci(level = 0.80)
## # A tibble: 1 × 2
## lower_ci upper_ci
## <dbl> <dbl>
## 1 0.633 0.8
Using the app, with 80% confidence level, it shows the true population to be lower than 95%. With the true population being lower, the confidence level also decreases and is narrower.
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.I used the app to run 90% confidence level. This confidence level, when compared to the first one is much wider. When calculating the bounds of the interval you get (61.70%, 80%).
#90% confidence level
%>%
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 0.617 0.817
As you increase the sample size, the width of the intervals decrease and vice versa.
Using the app I noticed that increasing the number of bootstrap will the sampling distribution narrow, therefore, it should decrease the standard error as well.