8.5: Tests for a Single Proportion

Many real-world questions boil down to a single idea: What proportion of people... agree with something, own something, choose something, or took some action?

In this section, we’ll explore how to test claims about population proportions — the fraction or percentage of a population that fits a category. These types of questions are everywhere: in surveys, polls, tracking forms, and policy planning.

Recall: What is a Proportion?

Remember that a proportion is a number between 0 and 1 that represents the part of a group that meets a condition. Proportions can also be shown as percentages (multiply by 100).

Here are a few examples:

• You survey 40 students, and 25 say they study off-campus. What’s the proportion?
• In a local election, 1,080 voters out of 1,500 said “yes” to a tax measure. What's the proportion who voted yes?
• 20 out of 25 light bulbs passed a quality test. What's the pass rate?

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What Questions Lead to a Proportion Test?

Here are five example research questions that naturally lead to testing a single proportion:

• Do more than 60% of local residents support building new bike lanes?
• Is customer satisfaction higher than 90% at a coffee shop chain?
• Are less than half of voters in favor of the proposed tax levy?
• Are more than 1 in 4 high school students getting less than 6 hours of sleep per night?
• Do more than 20% of all shoppers use coupons during checkout?

Each question can be approached by taking a sample, measuring the sample proportion, and comparing it to a stated or assumed population proportion using a z-test for proportions.

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Test Statistic for a Single Proportion

Here’s the formula we’ll use to test a null hypothesis involving a proportion:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

• \( \hat{p} \) = sample proportion
• \( p_0 \) = claimed (null hypothesis) population proportion
• \( n \) = sample size

This statistic measures how far away your sample result is from the null claim, in terms of standard error. We’ll use it just like with z-tests for means → to calculate a p-value and decide whether to reject \( H_0 \).

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Example: “Don’t you think most people want more parks?”

A city planner claims that most residents support adding more public parks. “Most,” you assume, should mean more than 50%.

You collect a random sample of 120 residents. Out of them, 69 say they support funding for more parks. Can we conclude that more than half of the population feels this way?

Let’s use our 5-step testing framework.

Step 1: State the Hypotheses

• \( H_0: p = 0.50 \) (half of the population supports it)
• \( H_A: p > 0.50 \) (more than half support more parks)

This is a one-tailed (right side) test.

Step 2: Select Test and Plug In

Your sample proportion is:

\( \hat{p} = \frac{69}{120} = 0.575 \)

Now compute your test statistic:

\[ Z = \frac{0.575 - 0.50}{\sqrt{\frac{0.50 \cdot (1 - 0.50)}{120}}} = \frac{0.075}{\sqrt{\frac{0.25}{120}}} = \frac{0.075}{0.0456} \approx 1.645 \]

Step 3: Choose Significance Level

We’ll use \( \alpha = 0.05 \)

Step 4: Find the p-value

Using technology or a z-table, \( P(z > 1.645) \approx 0.0500 \)

Step 5: Make Your Decision

Since \( p = 0.0500 = \alpha \), we are exactly on the boundary!

• Some instructors will say: reject if \( p \leq \alpha \). In that case → reject \( H_0 \)
• Others require \( p < \alpha \). In that case → fail to reject

Conclusion: This result is right on edge. There is borderline evidence that more than half of residents support more parks — but we can’t say so with high confidence.

Reflect: With more data, we might be able to know for sure — but in this sample, support seems just above 50%. It's worth noting that at a lower significance level we would conclude that there is not sufficent evidence for majority support of parks, but at a higher significance level there would be sufficient evidence for our conclusion.

This shows how the significance level is not set in stone and can influence our conclusions, which is why we need to set it before performing the study. How strong of evidence to we want to have before we make our conclusion? This is why interpreting a p-value is not as straight forward as some study authors might imply.

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