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?

A proportion is a number between 0 and 1 that represents the part of a group that meets a condition. Proportions can also be expressed 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 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 your sample result is from the null claim, in terms of standard error. We'll use it just like the z-tests for means — to calculate a p-value and decide whether to reject \( H_0 \).

Conditions for Using This Test

Before running a one-proportion z-test, verify that these conditions are satisfied:

• Randomness: The sample was collected using a random process.
• Independence: Individual observations are independent of one another. If sampling without replacement, the sample should be no more than 10% of the population.
• Success-failure condition: The sample must be large enough that both \( np_0 \geq 10 \) and \( n(1 - p_0) \geq 10 \). This ensures the sampling distribution of \( \hat{p} \) is approximately normal.

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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 five-step testing framework.

Step 1: State the Hypotheses

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

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

Step 2: Select Test and Calculate the Test Statistic

Your sample proportion is:

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

Check the success-failure condition: \( np_0 = 120(0.50) = 60 \geq 10 \) and \( n(1-p_0) = 60 \geq 10 \). ✓

Now compute the test statistic:

\[ Z = \frac{0.575 - 0.50}{\sqrt{\frac{0.50 \times 0.50}{120}}} = \frac{0.075}{\sqrt{0.002083}} = \frac{0.075}{0.04564} \approx 1.64 \]

Step 3: Choose a Significance Level

We'll use \( \alpha = 0.05 \).

Step 4: Find the p-value

Using technology or a z-table, \( P(Z > 1.64) \approx 0.0505 \).

Note on rounding: You may have seen the value 1.645 appear in z-tables as the critical value for a right-tailed test at exactly \( \alpha = 0.05 \). Our computed statistic of 1.64 is very close — which is why the p-value lands right at the boundary. This near-coincidence is worth noticing: when \( Z_{obs} \approx Z^* \), the p-value \( \approx \alpha \), and we are right on the edge of the decision threshold.

Step 5: Make Your Decision

Since \( p \approx 0.0505 > \alpha = 0.05 \), we fail to reject \( H_0 \), but just barely.

• Some instructors use \( p \leq \alpha \) as the rejection rule, in which case this result would reject \( H_0 \).
• Others require \( p < \alpha \) strictly, in which case we fail to reject.

Conclusion: This result is right on the edge. There is borderline evidence that more than half of residents support more parks, but we cannot say so with high confidence. It's important to be transparent about our results and always detail our methodologies and p-value. As you can see in these borderline cases, that rejection is fairly arbitrary as it is set by our \(\alpha\). The actual numbers and p-value give much more information.

Reflect: With more data, we might be able to resolve this either way, but in this sample, support appears just above 50%. It's worth noting that at a lower significance level we would conclude there is not sufficient 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 influences our conclusions, which is exactly why we set it before performing the study. How strong a standard of evidence do we want before drawing a conclusion? This is why interpreting a p-value is not as straightforward as some study authors imply.

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Example 2: "Are fewer than 30% of students completing the optional review?" (Try It)

An instructor believes fewer than 30% of students in a large introductory course complete the optional weekly review module. She randomly selects 85 students and finds that 21 completed the review last week.

Use the five-step framework to test her claim at \( \alpha = 0.05 \).

1. State \( H_0 \) and \( H_A \). What type of test is this — left, right, or two-tailed?
2. Calculate \( \hat{p} \) and check the success-failure condition using \( p_0 = 0.30 \).
3. Calculate the z test statistic.
4. Find the p-value and compare it to \( \alpha = 0.05 \).
5. Write a conclusion in context.

Hint: Since the instructor believes the proportion is less than 30%, this is a left-tailed test. The p-value will be \( P(Z < z_{obs}) \).

Tip: \( \hat{p} = 21/85 \approx 0.247 \). Use \( p_0 = 0.30 \) in the standard error formula, not \( \hat{p} \). The null value is always used in the denominator for a one-proportion z-test.

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Looking Ahead

Now that we can test a claim about a single proportion, the natural next question is: what if we want to compare two proportions from two different groups? In the next section, we'll extend this test to handle questions like "Is the pass rate higher in one program than another?" or "Do two age groups differ in their support for a policy?" The logic and the five-step framework stay exactly the same, only the test statistic changes.