Topic 6.4 Notes – Setting Up a Test for a Population Proportion

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

This test is used when you have:

• One categorical variable
• Each individual is classified as success/failure
• A claim about one population proportion $p$

Key pieces:

• Parameter: $p$ = true population proportion
• Statistic: $\hat{p} = \frac{x}{n}$ = sample proportion
• Claimed value: $p_0$

You assume the null hypothesis is true (that $p = p_0$) and ask:

If that were true, how unusual is our sample result?

Because of the normal model for sampling distributions, we can use a z-test when conditions are met.

This is not a t-test and not a chi-square test. If it’s one proportion and success/failure data, this is your method.

2. Writing Hypotheses for a Population Proportion

Everything revolves around the parameter $p$, not $\hat{p}$.

The Null Hypothesis $H_0$

• Always about the population proportion
• Always includes an equality

$$H_0: p = p_0$$

Even if the wording says “at least 40%,” we test at the boundary:

$$H_0: p = 0.40$$

The null represents the status quo or no change.

The Alternative Hypothesis $H_a$

• Also about $p$
• Uses a strict inequality

Three possibilities:

• $H_a: p > p_0$ (right-tailed)
• $H_a: p < p_0$ (left-tailed)
• $H_a: p \ne p_0$ (two-tailed)

The direction comes from the research question, not the data.

If the question asks, “Has the proportion increased?” → use $>$.
If it asks, “Is it different?” → use $\ne$.

Students often mess this up by choosing the direction based on $\hat{p}$. Don’t do that. The alternative is decided before looking at results.

Always define $p$ in context. For example:

$p$ = true proportion of seniors at Central High who plan to attend a four-year college.

That wording matters on FRQs.

3. Conditions for a One-Proportion z-Test

Before doing inference, you must justify that the normal model applies.

Random

Data must come from:

• A random sample, or
• A randomized experiment

You must say this in context. If it’s not random, you cannot generalize to the population.

Independence

Observations must be independent.

If sampling without replacement, check the 10% condition:

$$n \le 0.10N$$

In words:

The population is at least 10 times the sample size.

This prevents dependence from being a problem.

Large Counts Condition (Normality)

We model $\hat{p}$ with a normal distribution assuming $H_0$ is true.

That means we use $p_0$, not $\hat{p}$.

Check:

$$np_0 \ge 10$$

$$n(1 - p_0) \ge 10$$

Both expected successes and expected failures must be at least 10.

This is one of the most common AP mistakes. If you use $\hat{p}$ here, you lose credit.

Why the Normal Model Works

Here’s the idea visually. Under the null, the sampling distribution of $\hat{p}$ is centered at $p_0$ and has standard deviation $\sqrt{\frac{p_0(1-p_0)}{n}}$. The shaded right tail represents the probability of getting a sample proportion that far above $p_0$ if $H_0$ were true.

Under $H_0$, the sampling distribution of $\hat{p}$:

• Is approximately normal
• Has mean $p_0$
• Has standard deviation $\sqrt{\frac{p_0(1-p_0)}{n}}$

If your sample proportion falls far out in a tail, that’s evidence against $H_0$.

Choosing the Correct Test Method

When you see:

• One categorical variable
• A single sample
• A claim about one population proportion

The correct method is:

One-sample z-test for a population proportion

On multiple choice, they often describe the scenario in words and expect you to recognize this structure without saying “z-test.” Train yourself to scan for: single proportion + success/failure.