Topic 3.5 Notes – Setting Up a Test for a Population Proportion

What a One-Sample z-Test for a Population Proportion Is

This test is for one population proportion, written as $p$. That means you are asking about the true proportion in a population that has some outcome of interest.

You use it when all of these are true:

• there is one population
• there is one sample
• the response variable is categorical
• you can code responses as success/failure

A success is just the category you care about. It does not mean “good.”

• If the variable is “defective or not defective,” then “defective” can be success.
• If the variable has several categories, that still works if you combine them into two groups. Example: “supports” = success, and “neutral or opposes” = failure.

The parameter and statistic are easy to mix up:

• $p$ = the true population proportion
• $\hat p$ = the sample proportion, found by $$\hat p = \frac{x}{n}$$

Hypotheses are always about $p$, never about $\hat p$, because $\hat p$ is already known from your sample.

This is not the right procedure for:

• estimating a proportion with a confidence interval
• comparing two groups
• working with quantitative data

Writing the Parameter and Hypotheses

Your parameter definition needs three pieces:

• the population
• the success category
• that it is a population proportion

Example:

• $p$ = the true proportion of all students at Central High who prefer later school start times

The null hypothesis has the form

$$H_0: p = p_0$$

Here, $p_0$ is the claimed or benchmark proportion, and it must be between 0 and 1.

The alternative depends on the wording:

• lower-tailed $$H_a: p < p_0$$
• upper-tailed $$H_a: p > p_0$$
• two-sided $$H_a: p \ne p_0$$

Wording clues:

• less than, below, lower, decreased $\rightarrow <$
• greater than, above, higher, increased $\rightarrow >$
• different, changed, not equal $\rightarrow \ne$

One thing AP loves here. The null gets the equality. Even if the situation sounds like “at least 60%,” AP Stats writes the test as $H_0: p = 0.60$.

That last wording clue matches a two-sided test, where extreme results in either direction count against $H_0$.

Two-tailed rejection regions

How to Set Up the Test

This setup follows a clean order:

1. Identify the response variable.
2. Define what counts as success.
3. Name the parameter $p$ in context.
4. Decide whether this is a one-sample $z$-test for a population proportion.
5. Find the null value $p_0$ from the claim.
6. Write $H_0: p = p_0$.
7. Choose $H_a$ from the question wording, not from the sample result.
8. State whether the test is one-sided or two-sided.
9. Check conditions before doing any test statistic or p-value work.

Quick recognition checklist:

• one sample
• one population
• categorical response
• proportion of successes is the target

Conditions for Using the Test

All three conditions must be justified.

Randomization condition

The data should come from a random sample. You justify this from how the sample was selected, not by saying the sample size is large.

A big sample does not fix a biased method like a convenience sample.

10% condition

If sampling without replacement, you need

$$N \ge 10n$$

This is about approximate independence. It compares sample size to population size. It is not the same as “your sample has to be big.”

Normality condition

Use expected counts under the null, so use $p_0$, not $\hat p$:

$$np_0 \ge 10\qquad \text{and} \qquad n(1-p_0) \ge 10$$

You must check expected successes and expected failures separately.

That differs from a confidence interval, where the check uses the sample counts based on $\hat p$.

What Students Mix Up

These are the mistakes that show up all the time:

• Writing hypotheses with $\hat p$ instead of $p$
• Picking the direction of $H_a$ because the sample proportion ended up above or below $p_0$
• Forgetting to define success clearly
• Using this test for two groups or for quantitative data
• Using observed counts for normality instead of expected counts under $H_0$
• Taking $p_0$ from the sample instead of from the claim
• Saying “random” without saying how the sample was chosen
• Mixing up the 10% condition with the large-counts condition
• Trying to finish the whole test here when this topic only covers correct setup