Topic 6.5 Notes – Interpreting p-Values

1. What a p-Value Is

A p-value is the probability of getting a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis $H_0$ is true.

Two pieces are always built in:

• We assume the null hypothesis is true (for a proportion test, that means $p = p_0$).
• We look at how unusual our sample result is in the null distribution.

The null distribution can come from:

• A randomization distribution (simulation-based), or
• A theoretical distribution like the normal (z) distribution when conditions are met.

For a one-sample proportion test:

• We assume the true proportion is $p_0$.
• If $np_0 \ge 10$ and $n(1-p_0) \ge 10$, the sampling distribution of $\hat{p}$ is approximately normal.

So the p-value answers this question:

If the true population proportion were $p_0$, how likely is a sample result this extreme?

Small p-value → data are unlikely under $H_0$ → stronger evidence against $H_0$.
Large p-value → data are plausible under $H_0$ → weak evidence against $H_0$.

2. The z-Test Statistic for a Population Proportion

Every test statistic follows the same structure:

$$\text{test statistic} = \frac{\text{sample statistic} - \text{null value}}{\text{standard deviation under } H_0}$$

For a one-sample proportion:

$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$$

Where:

• $\hat{p}$ = sample proportion
• $p_0$ = hypothesized proportion from $H_0$
• $n$ = sample size

Notice the denominator uses $p_0$, not $\hat{p}$. That’s because we are building the distribution assuming the null is true.

The z-score tells you how many standard deviations $\hat{p}$ is from $p_0$:

• Positive z → $\hat{p} > p_0$
• Negative z → $\hat{p} < p_0$

Once you have z, you use the normal distribution to find the p-value.

3. Finding the p-Value from the z-Statistic

The p-value is the area in the tail(s) of the standard normal curve beyond your observed z. The direction depends on the alternative hypothesis.

Right-tailed test $H_a: p > p_0$

• p-value = area to the right of z
• We’re looking for unusually large sample proportions.

Left-tailed test $H_a: p < p_0$

• p-value = area to the left of z
• We’re looking for unusually small sample proportions.

Two-tailed test $H_a: p \ne p_0$

• p-value = area in both tails beyond ±|z|
• Double the one-tail area.

The graph below shows a left-tailed example, where the shaded region represents the area to the left of a given z-score.

Left-tailed p-value on the standard normal curve

On multiple-choice questions, a common trap is doubling the area when the alternative is one-sided. Always check the direction of $H_a$ before finding area.

4. Interpreting the p-Value in Context

Your interpretation must include:

• A probability statement
• The phrase “assuming the null hypothesis is true”
• The real-world context

A strong AP-style sentence sounds like this:

Assuming the true proportion of seniors at this school who plan to attend college is 0.65, the probability of obtaining a sample proportion as extreme or more extreme than the one observed is 0.028.

That’s it. Clear and tied to the actual population.

Connecting to significance level $\alpha$

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

Rejecting $H_0$ means the data provide convincing evidence for the alternative.
Failing to reject means the data do not provide convincing evidence.

It does not mean the null is true.

5. Common Mistakes

• Saying the p-value is the probability that $H_0$ is true. We assume $H_0$ is true when calculating it.
• Forgetting “as extreme or more extreme.” The p-value includes more unusual outcomes than what you observed.
• Using $\hat{p}$ in the standard deviation instead of $p_0$.
• Giving a generic interpretation with no context. On free-response, missing context costs points immediately.
• Writing “there is a 3% chance the results happened by random chance.” That’s vague. Tie it to the null model and the specific proportion.