STA 111 Lab 5

The Goal

In the last lab, we explored sampling variability of a sample proportion. We saw that different samples from the same population can yield different sample proportions. We also saw that we can use the sampling distribution as a way to estimate how much we expect a sample proportion to change if we took a different sample from the sample population. We discussed how examining and understanding the sampling variability allows us to report not just our sample proportion but a range of plausible values for the population proportion. Today we are going to formalize this concept with confidence intervals.

The Data

Gallup is an organization that conducts extensive polls aimed at exploring a variety of facets about societal opinions, political views, and more. On June 11, 2025, the Gallup organization released an article ( https://news.gallup.com/poll/691424/americans-religion-increasing-influence.aspx) that states that 34% of Americans believe that the influence of religion on society in the United States is increasing. The methods Gallup uses to obtain its data can be found here: https://www.gallup.com/175307/gallup-poll-social-series-methodology.aspx.

This article is an example of a news article that uses statistics. In this course, we have talked about the need to confirm the reliability of data before trusting any conclusions an article draws from that data. One thing to look for to assess the validity of supplied data is information like margins of error, sampling methods, and sample sizes. If this information is provided, this lends more support to the claim that the data presented can be safely used. It also allows us to assess any potential biases that may result from the data collection methods.

You may have to open the article and take a look to find the answers to the questions below. Pay special attention to the Survey Methods section at the end - most of the answers are there. A screen shot of this section is included below.

Question of Interest

Based on the survey results, we are able to determine what proportion of people in the sample reported religion as being very important. We have a researcher who wants to know if this suggests that less than 35% of all Americans feel the influence of religion on society in the United States is increasing.

Confidence Interval for a Population Proportion

Rather than relying on just the proportion reported in the article, we are going to use a confidence interval to answer the researcher’s question. Remember that a confidence interval is a range of plausible values for a population parameter.

Last time, we used simulations to explore sampling variability. Specifically, we built our sampling distribution of the sample proportion by drawing many samples from a population. However, we can’t do that today, as we do not have population data.

Luckily, we have a beautiful mathematical result that allows us to describe the sampling distribution of the sample proportion without needing to run a simulation. When certain conditions are met, we can assume that the sampling distribution of \(\hat{p}\) is a normal distribution with mean \(p\), and we can approximate the standard error (standard deviation) with

\[SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\] where \(\hat{p}\) is the sample proportion and \(n\) is the sample size.

Okay, so let’s use our normal distribution to describe our sampling distribution. In order to do that, we need to compute our standard error.

Once we have computed the standard error, we need to make sure we understand what it tells us. We have decided that a normal distribution can be used to describe the sampling distribution of \(\hat{p}\).

The value you have computed in Question 10 is called the margin of error (ME). Basically, if we start from our sample proportion \(\hat{p}\), and stretch out a distance equal to the margin of error above and below \(\hat{p}\), we will catch \(p\) in this interval for 95% of our samples.

We call the range of values that this distance spans a confidence interval.

Confidence Interval for \(p\)

\[\hat{p} \pm ME\]

# The Lower Bound 
phat - ME
# The Upper Bound 
phat + ME

Wrapping Up

Confidence levels are extremely powerful tools. They allow us to use just one sample to estimate a range of plausible values for a population proportion. We will learn to make confidence intervals for other parameters, like population means, as we move through this course.