Topic 6.6 Notes – Concluding a Test for a Population Proportion

1. What It Means to Conclude a Significance Test for a Population Proportion

At this point, you already have:

• Hypotheses (like $H_0: p = 0.40$, $H_a: p > 0.40$)
• A test statistic $z$
• A p-value

Now you must decide whether the sample provides convincing evidence against $H_0$.

Two ideas drive everything:

• The p-value is the probability of getting your result (or something more extreme) if the null hypothesis is true.
• The significance level $\alpha$ is your cutoff for deciding what counts as “small.”

Your job is to:

1. Compare p-value to $\alpha$
2. Make a decision about $H_0$
3. State a conclusion about the true population proportion $p$ in context

This is the moment where statistical reasoning answers the research question.

2. The Significance Level α and the Decision Rule

What α Means

The significance level $\alpha$ is:

$$\alpha = P(\text{reject } H_0 \mid H_0 \text{ is true})$$

That’s the probability of a Type I error.

If $\alpha = 0.05$, you are accepting a 5% chance of rejecting a true null hypothesis.

Important details:

• It is chosen before looking at the data.
• Common values are 0.05, 0.01, and 0.10.
• Smaller $\alpha$ means you require stronger evidence to reject $H_0$.

The Formal Decision Rule

You must explicitly compare the numbers:

• If p-value ≤ $\alpha$ → Reject $H_0$
• If p-value > $\alpha$ → Fail to reject $H_0$

That comparison must appear in writing.

Correct phrasing:

Since the p-value (0.018) is less than $\alpha = 0.05$, we reject $H_0$.

Never say:

• “Accept $H_0$”
• “Prove the null”
• “Accept the alternative”

You only reject or fail to reject.

3. What the p-Value Is Actually Telling You

It helps to picture this on a normal curve. The shaded tail represents the p-value, which is the probability of getting a result at least as extreme as your observed statistic, assuming $H_0$ is true.

For a right-tailed test like $H_a: p > 0.40$ with observed $z = 2.10$, the p-value is the dark red tail beyond 2.10 (about 0.018). It sits inside the $\alpha = 0.05$ region beyond 1.645, so p-value < $\alpha$ and we reject $H_0$.

Small p-values

A small p-value means:

• Your observed result would be unusual if $H_0$ were true.
• That gives evidence in favor of $H_a$.
• The smaller the p-value, the stronger the statistical evidence against $H_0$.

You’re basically saying:

“If the null were true, this result would be rare.”

Large (Not Small) p-values

A large p-value means:

• The result is not unusual under $H_0$.
• There is not convincing evidence for the alternative.

Crucial distinction:

• Failing to reject $H_0$ does not mean it is true.
• It means you do not have strong enough evidence to support $H_a$.

On AP free response, students lose points every year for writing something like “This proves the population proportion is 0.60.” That can never come from a significance test.

4. Writing the Full Conclusion in Context

Your conclusion has three required parts.

1. Compare p-value to $\alpha$

State the numbers clearly.

2. State the Decision

Reject or fail to reject $H_0$.

3. Interpret in Context About the True Population Proportion

You must reference:

• The population
• The variable
• The alternative hypothesis

Here’s the structure you should internalize:

Since the p-value (____) is [less than/greater than] $\alpha$ = ____, we [reject/fail to reject] $H_0$. There is [sufficient/insufficient] statistical evidence to conclude that [state $H_a$ in context about the true population proportion].

Example (structure only):

There is sufficient statistical evidence to conclude that the true proportion of high school students who prefer later start times is greater than 0.50.

Notice:

• It says true proportion
• It references the actual population
• It matches the alternative hypothesis exactly

The conclusion is your statistical justification for answering the research question.