6.6 Concluding a Test for a Population Proportion

You compare the p-value to your chosen significance level α. If p-value ≤ α, reject H0; if p-value > α, fail to reject H0 (CED DAT-3.B.2 and DAT-3.B.8). Rejecting H0 means there’s sufficient statistical evidence for Ha in context; failing to reject means there isn’t convincing evidence for Ha, but you never “prove” H0 true (CED DAT-3.B.3 and DAT-3.B.5). Also interpret in context (mention the population and the claim) and report the decision formally (e.g., “At α = 0.05, p = 0.03 ≤ 0.05, so we reject H0 and conclude there is evidence that p is different than p0”). Small p-values give stronger evidence for Ha (CED DAT-3.B.6); non-small p-values don’t support Ha (CED DAT-3.B.7). Make sure conditions for a one-proportion z-test are met before trusting the p-value. For a quick review see the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and try practice problems (https://library.fiveable.me/practice/ap-statistics).

Rejecting H0 and proving Ha true are not the same. In hypothesis testing you compare the p-value to your predetermined α. If p ≤ α you reject H0—that means the data provide sufficient statistical evidence for the alternative in context (CED DAT-3.B.2–B.3, B.4). It’s a statement of evidence, not a logical proof. A small p-value says the observed result would be unlikely if H0 were true, so the model under H0 is doubtful (CED DAT-3.B.6). You can never “prove” Ha is true with a significance test. Likewise, if p > α you fail to reject H0—that means there’s insufficient evidence for Ha, not that H0 is proven (CED DAT-3.B.5, B.7). Always state your conclusion in context on the AP exam. For more practice and wording examples, see the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and try problems at (https://library.fiveable.me/practice/ap-statistics).

Think of the p-value as how surprising your data are if the null hypothesis H0 (p = p0) were true. The formal decision rule you must use on the AP exam is simple and required by the CED: compare the p-value to the predetermined significance level α. - If p-value ≤ α, reject H0. (You have sufficient statistical evidence for Ha.) - If p-value > α, fail to reject H0. (You do NOT have sufficient evidence for Ha—you don’t “prove” H0 true.) Quick examples: p = 0.03 with α = 0.05 → 0.03 ≤ 0.05, so reject H0 and conclude in context that there’s evidence for Ha. p = 0.12 with α = 0.05 → 0.12 > 0.05, so fail to reject H0 and say there’s insufficient evidence for Ha. Always state your conclusion in context and note whether the test was one-sided or two-sided (this affects the p-value). For AP review, see the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and try practice problems (https://library.fiveable.me/practice/ap-statistics) to get comfortable with the comparisons.

For a one-proportion z-test the test statistic is z = (p̂ − p0) / sqrt[ p0(1 − p0) / n ], where p̂ is the sample proportion, p0 is the null value (H0: p = p0), and the standard error uses p0. Use a z-test only when the sample is random, independent (10% condition if sampling without replacement), and np0 and n(1−p0) are at least about 10. Compare the resulting z to a normal model to get the p-value and then compare the p-value to α to decide (reject if p-value ≤ α). The AP exam supplies formula/tables, but you should memorize this form and the conditions. For a quick review see the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and try practice problems (https://library.fiveable.me/practice/ap-statistics).

Use a one- or two-tailed test based on the research question (the claim), not the data. On the AP, your null is always H0: p = p0. Choose Ha to match the claim: - Ha: p > p0—use a right-tailed (one-sided) test when you’re testing “greater than” (e.g., “more than 50%”). - Ha: p < p0—use a left-tailed (one-sided) test when you’re testing “less than.” - Ha: p ≠ p0—use a two-tailed test when you’re testing “different from” or just “is not equal to.” Interpretation: compare the p-value (from the one-proportion z-test) to α. If p ≤ α, reject H0 and conclude in context that there’s evidence for Ha (DAT-3.B.2–B.4). Small p-values give evidence for the direction in Ha (DAT-3.B.6). Also verify inference conditions (random sample and np0 and n(1−p0) large enough)—see Topic 6.4 and the concluding-test study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g). For more practice, try problems at (https://library.fiveable.me/practice/ap-statistics).

Step-by-step: 1) State H0: p = p0 and Ha (one- or two-sided) in context. 2) Check conditions: random sample, independence (n ≤ 10% of pop), and np0 and n(1−p0) ≥ 10 so the sampling distribution of p̂ is approx. normal. 3) Compute p̂ = x/n. Use the one-proportion z-test: z = (p̂ − p0) / sqrt[p0(1−p0)/n]. (Note: standard error uses p0 under H0.) 4) Find the p-value from the standard normal (use right/left/two-tail based on Ha). 5) Formal decision: compare p-value to α. If p-value ≤ α, reject H0; if p-value > α, fail to reject H0 (CED DAT-3.B.2, B.8). 6) State conclusion in context: either “There is sufficient statistical evidence to support Ha” or “There is insufficient evidence to support Ha.” 7) Add caveats: you can reject H0 but never prove it true; small p-values give stronger evidence (CED DAT-3.B.3–B.7). For a clear walk-through and examples, see the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g). For more practice, check the Unit 6 overview (https://library.fiveable.me/ap-statistics/unit-6) and 1000+ practice questions (https://library.fiveable.me/practice/ap-statistics).

Write two sentences: a formal decision (compare p-value to α) and a conclusion in context that ties to the alternative hypothesis. 1) Formal decision: state the test outcome using CED language—e.g., "Because p-value ≤ α, reject H0: p = p0" or "Because p-value > α, fail to reject H0: p = p0." (Always compare the p-value to your chosen α.) 2) Contextual conclusion: translate that decision into plain context and link to Ha. If you rejected H0 say, "There is convincing statistical evidence that the population proportion is (greater than / less than / different from) p0," using the wording of Ha. If you fail to reject, say, "There is insufficient statistical evidence to conclude that the population proportion is (greater than / less than / different from) p0." Remind: failing to reject is not proof H0 is true (CED DAT-3.B.5). Want examples and quick practice? See the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g), the Unit 6 overview (https://library.fiveable.me/ap-statistics/unit-6), and lots of practice problems (https://library.fiveable.me/practice/ap-statistics).

A p-value of 0.003 means that if the null hypothesis (H0: p = p0) and your probability model were true, the chance of getting a sample result as extreme (or more) than what you observed is 0.3%—very unlikely. Because the p-value is much smaller than common significance levels (e.g., α = 0.05 or 0.01), you would reject H0 and say there is convincing statistical evidence in favor of the alternative hypothesis (Ha). Always state that conclusion in context (what proportion, which population, and what direction Ha claims). Remember: a small p-value is evidence against H0, not a proof that Ha is absolutely true, and it doesn’t measure practical importance. For AP exam practice and phrasing, see the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and try more problems at (https://library.fiveable.me/practice/ap-statistics).

We say “fail to reject H0” instead of “accept H0” because a significance test only tells you whether there’s enough evidence against the null—it can’t prove the null is true. If p-value ≤ α, you reject H0; if p-value > α, you fail to reject H0 (CED DAT-3.B.2, DAT-3.B.3, DAT-3.B.5). A large p-value means the observed result isn’t unusual under H0, so you don’t have convincing evidence for Ha, but that doesn’t prove H0. Saying “accept” would overstate the result and ignore possibilities like low power or a Type II error (you might miss a real effect). Always state your conclusion in context and compare p to α (CED DAT-3.B.4, DAT-3.B.8). For a quick refresher on wording and examples, see the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and try practice problems (https://library.fiveable.me/practice/ap-statistics).

Do a one-proportion z-test in 5 clear steps so you won’t lose points on the exam: 1) State H0 and Ha in context (H0: p = p0; Ha: p <, >, or ≠ p0). 2) Check conditions: random sample, independence (n ≤ 10% of pop), and np0 and n(1−p0) ≥ 10 so the sampling distribution is approx. normal. 3) Compute p̂ = x/n, standard error SE = sqrt(p0(1−p0)/n), and test statistic z = (p̂ − p0)/SE. 4) Find the p-value from z (use your calculator/table). Compare p-value to α: if p-value ≤ α, REJECT H0; if p-value > α, FAIL TO REJECT H0 (CED DAT-3.B.2, B.8). 5) State your conclusion in context: e.g., “There is sufficient statistical evidence at α = 0.05 to support the claim that …” or “There is insufficient evidence to conclude …” and remind that failing to reject H0 doesn’t prove H0 true (CED DAT-3.B.3, B.5). For a focused review, see the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and practice lots of problems (https://library.fiveable.me/practice/ap-statistics).

Alpha (α) is the significance level you choose before the test—the predetermined probability of rejecting H0 when H0 is actually true (a Type I error). For example, α = 0.05 means you’re willing to accept a 5% chance of a false alarm. The p-value is computed from your sample: it’s the probability, assuming H0 is true, of getting a test statistic as extreme (or more) than what you observed. The formal decision on the AP exam compares the two: if p ≤ α, reject H0; if p > α, fail to reject H0 (CED DAT-3.B.1–B.3, B.8). Small p-values give evidence for the alternative; large p-values don’t prove H0 true—they just say there’s insufficient evidence for Ha (CED DAT-3.B.6–B.7, B.5). For more on concluding tests for proportions, check the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and practice problems (https://library.fiveable.me/practice/ap-statistics).

Because 0.08 > 0.05, you fail to reject the null hypothesis. Formally: p-value (0.08) > α (0.05) ⇒ do not reject H0 (CED DAT-3.B.2, DAT-3.B.8). In context, this means the sample data do not provide convincing statistical evidence to support the alternative hypothesis; they’re not unusual enough under H0 to justify rejection. Important: “Fail to reject H0” is not the same as proving H0 true (CED DAT-3.B.5). It just means there’s insufficient evidence for the alternative at the 5% significance level. If you worry about missing a real effect, consider power or a larger sample (Type II error). For a refresher on wording and examples, see the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and try practice problems (https://library.fiveable.me/practice/ap-statistics).

You can’t ever prove H0 true because hypothesis testing is asymmetric: tests are designed to detect evidence against the null, not to confirm it. A p-value measures how unlikely the sample would be if H0 were true; a small p-value (≤ α) gives evidence to reject H0 and support Ha (CED DAT-3.B.6). But a large p-value only means the data aren’t unusual under H0—it doesn’t prove H0 is the real value (CED DAT-3.B.7, DAT-3.B.5). Practically, failing to reject H0 could be due to small sample size or low power (Type II error), so you never “accept” H0, you just “fail to reject” it. Remember the formal decision rule on the exam: compare p-value to α—if p ≤ α reject H0; if p > α fail to reject H0 (CED DAT-3.B.2, DAT-3.B.8). For the AP review on this topic see the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and practice problems (https://library.fiveable.me/practice/ap-statistics).

Statistical significance vs. practical significance—short answer: statistical significance tells you whether an observed difference in a proportion is unlikely under H0 (based on p-value and α); practical significance asks whether that difference matters in the real world. - Statistical significance (CED DAT-3.B): you compute a test statistic (one-proportion z), get a p-value, and compare to α. If p ≤ α you reject H0—you have sufficient statistical evidence for Ha. Always state this conclusion in context (CED DAT-3.B.4). - Practical significance: look at the size of the effect (p̂ − p0) and ask if it’s meaningful. A tiny difference (e.g., p0 = 0.50, p̂ = 0.52) can be statistically significant with a huge sample but may be trivial for decisions or policy. Tips: always report both—the formal p-value/decision and the actual effect size with context (and CIs help). Remember sample size affects statistical significance and Type I/II tradeoffs. For a quick refresher see the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and practice problems (https://library.fiveable.me/practice/ap-statistics).

You have enough evidence to support the alternative hypothesis when your test produces a p-value ≤ your chosen significance level α. Practically: (1) State H₀ and Hₐ in context (e.g., H₀: p = p₀, Hₐ: p > p₀). (2) Calculate the one-proportion z test statistic and p-value. (3) Compare p-value to α (α was set before you saw the data). If p-value ≤ α, reject H₀ and conclude there is sufficient statistical evidence for Hₐ (say that in context). If p-value > α, fail to reject H₀—there’s insufficient statistical evidence for Hₐ (not proof H₀ is true). Remember: smaller p-values give stronger evidence for Hₐ (CED DAT-3.B.6–B.9). Want a quick walkthrough and practice problems for this topic? Check the Topic 6.6 study guide (https://library.fiveable.me/ap-statistics/unit-6/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g) and try more practice questions (https://library.fiveable.me/practice/ap-statistics).