In August of 2012, news outlets ranging from the Washington Post to the Huffington Post ran a story about the rise of atheism in America. The source for the story was a poll that asked people, “Irrespective of whether you attend a place of worship or not, would you say you are a religious person, not a religious person or a convinced atheist?” This type of question, which asks people to classify themselves in one way or another, is common in polling and generates categorical data. In this lab we take a look at the atheism survey and explore what’s at play when making inference about population proportions using categorical data.
In this lab we will explore the data using the dplyr package and visualize it using the ggplot2 package for data visualization. The data can be found in the companion package for OpenIntro labs, oilabs.
Let’s load the packages.
library(tidyverse)
library(oilabs)
library(infer)The press release for the poll, conducted by WIN-Gallup International, can be accessed here.
Take a moment to review the report then address the following questions.
In the first paragraph, several key findings are reported. Do these percentages appear to be sample statistics (derived from the data sample) or population parameters? Explain your reasoning.
The title of the report is “Global Index of Religiosity and Atheism”. To generalize the report’s findings to the global human population, what must we assume about the sampling method? Does that seem like a reasonable assumption?
Turn your attention to Table 6 (pages 15 and 16), which reports the sample size and response percentages for all 57 countries. While this is a useful format to summarize the data, we will base our analysis on the original data set of individual responses to the survey. Load this data set into R with the following command.
data(atheism)atheism correspond to?To investigate the link between these two ways of organizing this data, take a look at the estimated proportion of atheists in the United States. Towards the bottom of Table 6, we see that this is 5%. We should be able to come to the same number using the atheism data.
us12 that contains only the rows in atheism associated with respondents to the 2012 survey from the United States. Next, calculate the proportion of atheist responses. Does it agree with the percentage in Table 6? If not, why?us12 <- atheism %>%
filter(nationality == "United States", year == "2012")As was hinted earlier, Table 6 provides statistics, that is, calculations made from the sample of 51,927 people. What we’d like, though, is insight into the population parameters. You answer the question, “What proportion of people in your sample reported being atheists?” with a statistic; while the question “What proportion of people on earth would report being atheists” is answered with an estimate of the parameter.
The traditional method of calculating a confidence interval approximates the sampling distribution by using the Normal distribution.
If the conditions for inference are reasonable, we can trust that the Normal curve provides a reasonably good approximation. Let’s proceed with calculating a traditional 95% confidence interval.
n <-
p_hat <-
SE <-
z_star <- qnorm( )
MoE <-
c(p_hat - MoE, p_hat + MoE)z_star is the number of standard errors that we’ll be going out to capture 95% of the sampling distribution. This number should be around 1.96, but you can be more precise by using the qnorm() function.We’ve seen how the bootstrap can provide an alternate path to estimating the standard error needed for our confidence interval. Let’s compare to see how different the two methods are.
Use infer to calculate the a bootstrap estimate of the standard error (that is, the standard deviation of the bootstrap distribution). Use this estimate to recompute the margin of error and the resulting confidence interval. Was it the same as or similar to your Normal theory approach?
Using both the Normal approximation approach and the bootstrap approach, calculate confidence intervals for the proportion of atheists in 2012 in one other country of your choice. Be sure to note whether the conditions for inference are met, and interpet the interval in context of the data.
Imagine you’ve set out to survey 1000 people on two questions: are you female? and are you left-handed? Since both of these sample proportions were calculated from the same sample size, they should have the same margin of error, right? Wrong! While the margin of error does change with sample size, it is also affected by the proportion.
Think back to the formula for the standard error: \(SE = \sqrt{p(1-p)/n}\). This is then used in the formula for the margin of error for a 95% confidence interval: \(ME = 1.96\times SE = 1.96\times\sqrt{p(1-p)/n}\). Since the population proportion \(p\) is in this \(ME\) formula, it should make sense that the margin of error is in some way dependent on the population proportion. We can visualize this relationship by creating a plot of \(ME\) vs. \(p\).
Since sample size is irrelevant to this discussion, let’s just set it to some value (\(n = 1000\)) and use this value in the following calculations:
n <- 1000The first step is to make a variable p that is a sequence from 0 to 1 with each number incremented by 0.01. We can then create a vector of the margin of error (me) associated with each of these values of p using the familiar approximate formula (\(ME = 2 \times SE\)).
p <- seq(0, 1, 0.01)
me <- 2 * sqrt(p * (1 - p)/n)Lastly, we plot the two vectors against each other to reveal their relationship. To do so, we need to first put these variables in a data frame that we can call in the ggplot function.
dd <- data.frame(p = p, me = me)ggplot to visualize the relationship between the Population Proportion and the Margin of Error (You can use either geom_point or geom_path and please add axis labels. Based on this plot, describe the relationship between p and me. For a given sample size, for which value of p is margin of error maximized?The question of atheism was asked by WIN-Gallup International in a similar survey that was conducted in 2005. (We assume here that sample sizes have remained the same.) Table 4 on page 13 of the report summarizes survey results from 2005 and 2012 for 39 countries. For the following questions, you may use either the Normal theory approach or the computational approach.
Is there convincing evidence that the US has seen a change in its atheism index between 2005 and 2012? As always, write out the hypotheses for any tests you conduct and outline the status of the conditions for inference. If you find a significant difference, also quantify this difference with a confidence interval.
If in fact there has been no change in the atheism index in the countries listed in Table 4, in how many of those countries would you expect to detect a change (at a significance level of 0.05) simply by chance?
Hint: Review the definition of the Type 1 error.
Suppose you’re hired by the local government to estimate the proportion of residents that attend a religious service on a weekly basis. According to the guidelines, the estimate must have a margin of error no greater than 1% with 95% confidence. You have no idea what to expect for \(p\). How many people would you have to sample to ensure that you are within the guidelines?
Hint: Refer to your plot of the relationship between \(p\) and margin of error. This question does not require using the dataset.