AP Statistics (AP Stats): Hypothesis Test for One Proportion (z?test for p)

AP Statistics – Hypothesis Test for One Proportion (z?test for p)

AP Statistics: Hypothesis Test for One Proportion (z-test for p) – Exam-Ready Study Guide

What This Is

A hypothesis test for one proportion determines whether a sample proportion (p?) provides convincing evidence that a population proportion (p) differs from a claimed value (p?). This is essential on the AP exam because it’s a foundational inference procedure, often tested in Free-Response Questions (FRQs) and multiple-choice. Real-world applications include: - Testing if a new drug’s success rate exceeds the current standard (e.g., does a vaccine work better than 70% efficacy?). - Determining if a factory’s defect rate is higher than advertised (e.g., is the proportion of faulty lightbulbs > 5%?). - Evaluating if a political candidate’s support has changed since the last poll.

Key Terms & Formulas

• Null Hypothesis (H?): The default claim; always written as H?: p = p?, where p is the true population proportion and p? is the hypothesized value.
• Alternative Hypothesis (H?): The claim we’re testing; can be one-sided (H?: p > p? or H?: p < p?) or two-sided (H?: p-p?).
• Sample Proportion (p?): p? = (number of successes) / n; the observed proportion in the sample.
• Test Statistic (z): z = (p? – p?) / ?(p?(1–p?)/n); measures how far p? is from p? in standard errors.
• P-value: Probability of observing a test statistic as extreme as (or more extreme than) the one calculated, assuming H? is true. Found using normalcdf(lower, upper, 0, 1) on TI-84.
• For H?: p > p?, use normalcdf(z, 1E99, 0, 1).
• For H?: p < p?, use normalcdf(-1E99, z, 0, 1).
• For H?: p-p?, use 2 * normalcdf(|z|, 1E99, 0, 1) (double the tail area).
• Significance Level (?): Threshold for rejecting H? (commonly-= 0.05). If p-value-?, reject H?.
• Conditions for Inference (BINS):
• Binary: Data are yes/no (success/failure).
• Independent: Sampled observations are independent (check 10% condition: n-0.10N, where N is the population size).
• Normal: Sampling distribution of p? is approximately normal (check Large Counts: np?-10 and n(1–p?)-10).
• Simple Random Sample (SRS): Data come from a random sample or randomized experiment.
• Calculator Command for Test: 1-PropZTest(p?, x, n, H?) where:
• p? = hypothesized proportion,
• x = number of successes,
• n = sample size,
• H? = alternative hypothesis (?p?, <p?, or >p?).

Step-by-Step / Process Flow

Follow these steps for every one-proportion z-test on the AP exam:

1. State the Hypotheses

2. Write H? and H? in context (define p clearly).

   • Example: p = true proportion of voters who support a policy.
   • H?: p = 0.60 (claimed support rate)
   • H?: p > 0.60 (testing if support has increased).

3. Check Conditions (BINS)

4. Binary? Yes/no data? (e.g., "support" vs. "don’t support").
5. Independent? Check 10% condition: n-0.10N.
6. Normal? Verify np?-10 and n(1–p?)-10.

7. SRS? Assume random sampling unless stated otherwise.

8. Calculate the Test Statistic

9. Use the formula z = (p? – p?) / ?(p?(1–p?)/n).

10. OR use 1-PropZTest on TI-84 (faster and less error-prone).

11. Find the P-value

12. Use normalcdf or the output from 1-PropZTest.

13. For two-sided tests, double the tail area.

14. Make a Conclusion in Context

15. Compare p-value to? (usually 0.05).
    • If p-value-?: "Reject H?. There is convincing evidence that [H? in context]."
    • If p-value > ?: "Fail to reject H?. There is not convincing evidence that [H? in context]."

16. Never say "accept H?"—only "fail to reject."

17. Interpret the P-value (if asked)

18. Example: "Assuming the true proportion of voters who support the policy is 60%, there is a [p-value] probability of getting a sample proportion of [p?] or more extreme by chance alone."

Common Mistakes

• Mistake: Forgetting to check the 10% condition when sampling without replacement.

• Correction: Always verify n-0.10N (even if the problem doesn’t mention it). The 10% condition ensures independence.

• Mistake: Using the sample proportion (p?) instead of p? in the standard error formula.

• Correction: The standard error is ?(p?(1–p?)/n), not ?(p?(1–p?)/n). The null hypothesis assumes p = p?.

• Mistake: Misinterpreting the p-value as the probability that H? is true.

• Correction: The p-value is the probability of the observed data (or more extreme), assuming H? is true. It does not measure the probability of H? itself.

• Mistake: Skipping the Normal condition check for small samples.

• Correction: If np? < 10 or n(1–p?) < 10, the sampling distribution of p? is not normal, and the z-test is invalid.

• Mistake: Writing conclusions without context.

• Correction: Always reference the real-world scenario (e.g., "There is convincing evidence that the drug’s success rate exceeds 80%").

AP Exam Insights

• FRQ Hotspots:
• You’ll often be given a real-world scenario (e.g., polling, medical trials) and asked to:
  1. State hypotheses in context.
  2. Check conditions (BINS).
  3. Calculate the test statistic and p-value (or use 1-PropZTest).
  4. Make a conclusion in context.

• Partial credit is given for correct hypotheses and conditions, even if the test statistic is wrong.

• Tricky Distinctions:

• z-test vs. t-test: Use a z-test for proportions (this topic) and a t-test for means. The AP exam rarely tests one-proportion t-tests.
• One-sided vs. two-sided tests: The p-value calculation differs (one tail vs. two tails). Read the question carefully!

• Confidence intervals vs. hypothesis tests: A CI estimates p, while a hypothesis test tests a claim about p. They’re related but not the same.

• Calculator Pitfalls:

• 1-PropZTest vs. 1-PropZInt: Don’t confuse the test (1-PropZTest) with the confidence interval (1-PropZInt). The exam may ask for one or both.
• Input errors: Double-check x (number of successes) and n (sample size). Mixing them up is a common mistake.

• Two-sided p-values: The calculator gives the one-tailed p-value for ?p?. You must double it for the correct two-sided p-value.

• Common FRQ Setups:

• "Is there evidence that the proportion has increased/decreased?"-One-sided test.
• "Has the proportion changed?"-Two-sided test.
• "Do the data provide convincing evidence...?"-Hypothesis test required.

Quick Check Questions

1. Multiple Choice: A factory claims that 5% of its lightbulbs are defective. In a random sample of 200 bulbs, 16 are defective. Which of the following is the correct test statistic for testing H?: p = 0.05 vs. H?: p > 0.05? (A) z = (0.08 – 0.05) / ?(0.05(0.95)/200) (B) z = (0.08 – 0.05) / ?(0.08(0.92)/200) (C) t = (0.08 – 0.05) / ?(0.05(0.95)/200) (D) z = (0.05 – 0.08) / ?(0.05(0.95)/200)

Answer: (A) Explanation: The standard error uses p? (0.05), not p? (0.08), and this is a z-test for proportions (not a t-test).*

1. FRQ Part: A researcher tests whether a new teaching method improves pass rates. Historically, 70% of students pass. In a random sample of 50 students using the new method, 40 pass.
2. a. State the hypotheses for this test.
3. b. Are the conditions for inference met? Justify your answer.
4. c. Calculate the test statistic and p-value. Show your work.

Answer: - a. H?: p = 0.70 (true pass rate with new method = historical rate) H?: p > 0.70 (true pass rate with new method > historical rate) - b. Conditions: - Binary: Pass/fail data (yes). - Independent: 50-0.10N (assume N-500, so yes). - Normal: np? = 50(0.70) = 35-10; n(1–p?) = 50(0.30) = 15-10 (yes). - SRS: Random sample (assumed). - c. p? = 40/50 = 0.80 z = (0.80 – 0.70) / ?(0.70(0.30)/50)-1.54 p-value = normalcdf(1.54, 1E99, 0, 1)-0.0618

Last-Minute Cram Sheet

1. Hypotheses: H?: p = p?; H?: p > p?, p < p?, or p-p? (define p in context!).
2. Test Statistic: z = (p? – p?) / ?(p?(1–p?)/n).
3. Conditions: BINS (Binary, Independent (10% condition), Normal (np?-10 and n(1–p?)-10), SRS).
4. P-value: Use normalcdf or 1-PropZTest (double for two-sided).
5. Conclusion: "Reject/fail to reject H?" + context.
6. Calculator: 1-PropZTest(p?, x, n, H?) (x = successes, n = sample size).
7. Always check the 10% condition (even if the problem doesn’t mention it).
8. Use p? (not p?) in the standard error for the test statistic.
9. Two-sided p-value = 2 × one-sided p-value (from 1-PropZTest).
10. Never say "accept H?"—only "fail to reject."