3.7 Carrying Out a Test for a Population Proportion

Concluding a one-proportion $z$ test comes down to comparing your p-value to your significance level, $\alpha$. If the p-value is less than or equal to $\alpha$, reject the null hypothesis; if it is greater than $\alpha$, fail to reject the null hypothesis.

Unit 3 (Inference for Categorical Data: Proportions) makes up about 25-35% of the multiple-choice section, and the conclusion step is where a full significance test turns numbers into a decision about the population.

Comparing the p-Value to Alpha to Decide

• The decision rule is simple: if p-value ≤ α, reject H₀; if p-value > α, fail to reject H₀.
• "Fail to reject" is never "accept." You can never prove the null hypothesis is true.
• A small p-value gives evidence for the alternative hypothesis. The smaller the p-value, the stronger that evidence.
• Rejecting H₀ means there is sufficient evidence for Hₐ; failing to reject means there is insufficient evidence for Hₐ.
• Every conclusion must be stated in context and refer to the true population proportion.
• The significance level α is the probability of rejecting H₀ when it is actually true.

The Decision Rule

After your hypotheses are written, conditions are checked, and your test statistic and p-value are calculated, you land on one of two outcomes: reject the null or fail to reject the null. Both come from comparing your p-value to the significance level (α), the predetermined cutoff you agreed to before collecting data.

• If the p-value ≤ α (often 0.05), reject H₀.
• If the p-value > α, fail to reject H₀.

Never write "accept H₀" or "accept Hₐ." Always use "reject" or "fail to reject." Failing to reject does not mean the null is true. It only means you do not have enough evidence to reject it.

Reasoning Through the p-Value

The p-value is the probability of getting your sample result, or something more extreme, assuming the null hypothesis and the probability model are true. You compare it directly to α.

If the p-value is below α, the observed result would be unusual under H₀, so you have reason to reject H₀ in favor of Hₐ. If the p-value is not below α, the result would not be surprising under H₀, so you fail to reject H₀. In that case you have neither strong evidence for Hₐ nor proof that H₀ is true.

Two ideas to keep straight:

• Small p-values mean the observed test statistic would be unusual if H₀ and the probability model were true, so they provide evidence for the alternative. The lower the p-value, the more convincing the evidence for Hₐ.
• p-values that are not small mean the observed test statistic would not be unusual under H₀, so they do not provide convincing evidence for Hₐ, and they do not provide evidence that H₀ is true.

The p-value measures evidence against the null. It does not give the probability that the alternative is true.

Using the z-Statistic to Reason

The p-value is the most common way to conclude, but the test statistic itself tells a similar story. A z-score measures how many standard deviations a value is above or below the mean. Because you already checked the normal condition, you are working with an approximately normal sampling distribution. When H₀ is true, the z-statistic z = (p̂ − p₀)/√(p₀(1−p₀)/n) follows approximately a standard normal distribution. This standard normal model is the probability model used to approximate the null distribution.

By the empirical rule, about 95% of a normal distribution falls within 2 standard deviations of the mean. So a z-statistic beyond 2 (or below -2) corresponds to a tail probability of roughly less than 0.05. A very large z (like 4 or more in absolute value) makes a reject H₀ decision clear. A z between -2 and 2 usually leads to fail to reject unless you have more information.

This is a quick reasoning check, not a replacement for actually computing the p-value and comparing it to α.

Writing the Conclusion to a Proportion Test

Free Response: When a test is part of an FRQ, your conclusion sentence is where points are won or lost. A reliable template:

"Since the p-value (value) is </> α (value), we reject / fail to reject H₀. We have / do not have convincing evidence of ________ (Hₐ in context)."

The three things to include every time:

1. Compare the p-value to the significance level.
2. State a decision: reject or fail to reject.
3. Write the conclusion in context, referring to the true population proportion.

A conclusion that skips the comparison, or that states a decision without context, is incomplete. Linking your numerical result to the research question is the point of the whole test.

Common Trap: Writing "accept the null" or treating a large p-value as proof that H₀ is true. Failing to reject only means you lack sufficient evidence for the alternative. Also avoid stopping at "reject H₀" without the in-context sentence about the population proportion.

MCQ: Multiple-choice questions often hand you a p-value and an α and ask for the correct decision, or give you a decision and ask what it means. Know that p-value ≤ α leads to rejecting, and that "fail to reject" never equals "the null is true."

Worked Example

A survey tests whether a new advertising campaign increases brand awareness. The null hypothesis is that the campaign has no effect on awareness; the alternative is that it increases awareness.

Before the campaign, 50% of people were aware of the brand, so $H_0: p = 0.5$ and $H_a: p > 0.5$, where $p$ is the true proportion of people aware of the brand after the campaign. A random sample of n = 400 people is taken after the campaign, and 57% are aware, so p̂ = 0.57. The test is run at α = 0.05.

The test statistic is $z = \frac{0.57 - 0.5}{\sqrt{0.5(0.5)/400}} = 2.8$.

What is the p-value, and what do you conclude?

The p-value is about 0.0026, meaning there is roughly a 0.26% chance of getting a test statistic as large as 2.8 if the null hypothesis is true. Since 0.0026 < 0.05, you reject the null hypothesis. There is convincing evidence that the true proportion of people aware of the brand after the campaign is greater than 0.5.

Common Misconceptions

• Misconception: Fail to reject means accept the null.
  Actually: It does not. You only conclude there is not enough evidence for the alternative. A test can never prove the null is true.
• Misconception: A large p-value proves H₀.
  Actually: A p-value that is not small just means your result would not be unusual under H₀. That is not evidence for H₀.
• Misconception: The p-value is the probability the null is true.
  Actually: The p-value assumes the null is true and measures how extreme your data are under that assumption.
• Misconception: "Reject H₀" is a complete answer.
  Actually: You still need the in-context sentence about the true population proportion. Skipping context costs credit.
• Misconception: Statistically significant means it matters in real life.
  Actually: Statistical significance and practical importance are not the same thing. A tiny effect can be significant with a large enough sample.
• Misconception: Choosing the wrong tail is appropriate.
  Actually: Make sure your p-value matches the alternative. A two-sided alternative uses both tails, so the p-value is 2·P(Z ≥ |z|). In a one-proportion z-test, you find the p-value from the standard normal distribution, using either a table or technology.

Related AP Statistics Guides

• Unit 3 Review: Inference for Categorical Data: Proportions
• 3.3 Constructing a Confidence Interval for a Population Proportion
• 3.5 Setting Up a Test for a Population Proportion
• 3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion
• 3.6 p-Values