Welcome back to Professor Baker's Math Class! In Section 6-3: The Statistics of Polling, we move beyond simple averages to answer a critical question in our data-driven world: Can we believe the polls? Whether it is a political election, a marketing survey, or a medical study, understanding the mechanics behind these numbers is essential for Quantitative Literacy.

Key Concepts: The Language of Polling

To understand survey results, we first need to master three specific terms that describe the reliability of the data:

  • Margin of Error: This expresses how close the poll result is expected to be to the true result of the entire population. It essentially tells us the "wiggle room" for the data.
  • Confidence Interval: This is the range of values derived by adding and subtracting the margin of error from the poll result. We expect the true population value to fall somewhere inside this range.
  • Confidence Level: This tells us how sure we can be. In this course, we primarily use a 95% Confidence Level, meaning if we conducted the same poll 100 times, we expect 95 of those results to contain the true population percentage.

The Mathematics of Accuracy

How do statisticians know how many people to interview? It turns out there is a direct mathematical relationship between the number of people surveyed ($n$) and the margin of error ($m$). Assuming a 95% confidence level, we use the following formulas:

1. Finding the Margin of Error
If you know how many people were surveyed ($n$), you can estimate the margin of error:

$$ \text{Margin of Error} \approx \frac{100}{\sqrt{n}} \% $$

2. Finding the Sample Size
If you know the desired margin of error ($m$) and want to know how many people to interview:

$$ \text{Sample Size} (n) \approx \left( \frac{100}{m} \right)^2 $$

Real-World Example

Let's look at an example from the class notes involving a survey of 1,294 residents regarding the disruption of their lives following Hurricane Katrina. If the poll reports that 41% of residents still feel disrupted, how accurate is that number?

First, we calculate the margin of error using our sample size ($n = 1294$):

$$ \frac{100}{\sqrt{1294}} \approx 2.8\% $$

Next, we construct the Confidence Interval by adding and subtracting this margin from the poll result (41%):

  • $41\% - 2.8\% = 38.2\%$
  • $41\% + 2.8\% = 43.8\%$

Conclusion: We can be 95% confident that the true percentage of residents whose lives are still disrupted lies somewhere between 38.2% and 43.8%.

Next Steps

Being able to think between the lines of statistical claims is a superpower. Review the full slide deck attached below for more examples, including how to handle margins of error in political approval ratings. Once you have reviewed the notes and the video lecture, you are ready to take the Section 6-3 Quiz!