Have you ever wondered how a news organization can survey just 1,000 people and claim to know what millions of Americans are thinking? In this lesson for Section 6-3: The Statistics of Polling, we answer the fundamental question: Can we believe the polls? Statistics allows us to "read between the lines" of raw data, but to do so effectively, we need to understand three core concepts: the margin of error, the confidence interval, and the confidence level.

Key Concepts in Polling

When analyzing the results of a survey or poll, keep these definitions in mind:

  • Margin of Error: This expresses how close the poll result is expected to be to the true result of the total population. It is often seen as the "plus or minus" figure.
  • Confidence Interval: This is the range of likely values. You find this by adding and subtracting the margin of error from the poll result.
  • Confidence Level: This tells us how sure we are. A 95% confidence level means that if we conducted the same poll 100 times, the results would fall within our confidence interval 95 times.

The Mathematics of Accuracy

Professor Baker wants you to be comfortable with the numbers behind the news. For a standard 95% level of confidence, we can estimate the relationship between the sample size and the margin of error using simple formulas.

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

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

For example, if you survey 900 people, your margin of error is approximately $\frac{100}{\sqrt{900}} = \frac{100}{30} = 3.3\%$.

2. Finding the Required Sample Size
Conversely, if you want a specific margin of error (represented by $m$), you can calculate how many people you need to interview:

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

As you can see from the formula, to cut your margin of error in half, you generally need to quadruple your sample size!

Interpreting the Results

It is crucial to look at the Confidence Interval rather than just the headline number. As discussed in the lecture notes regarding the Hurricane Katrina disruption poll:

If a poll shows 52% of people agree with a statement, but the margin of error is 2.8%, the true value lies between $49.2\%$ and $54.8\%$. Because the bottom of that interval drops below 50%, we cannot claim with absolute certainty that a "majority" agrees, even though the poll number is above 50%.

Watch Out for Bias!

Finally, as you complete the attached assignment, pay close attention to the wording of survey questions. Statistical math assumes a random sample and neutral questions. Questions that are phrased to encourage a specific answer (like the cigarette tax examples in your homework) introduce bias that no amount of math can fix.

Please review the attached PDF notes for detailed examples and complete the Section 6-3 assignment to test your skills.