AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/3/3-5)
Unit 3 · Topic 3.5
3.5 Setting Up a Test for a Population Proportion
A significance test asks whether sample data give convincing evidence against a claimed value of p. Setting one up means naming the test, defining the parameter, writing H₀ and Hₐ correctly and checking conditions, using the null value p₀.
Key terms
- null hypothesis (H₀)
- alternative hypothesis (Hₐ)
- one-sided vs. two-sided test
- one-sample z-test
- conditions for a test
The logic of a test
A test starts by assuming the null hypothesis H₀ is true. H₀ is the status quo or "no change" claim. The alternative hypothesis Hₐ is what you're looking for evidence of, usually the researcher's claim or suspicion.
You then ask: if H₀ were true, how surprising would my sample be? If it would be very surprising, that's evidence for Hₐ. This is like a trial where the defendant is presumed innocent (H₀) until the evidence is convincing.
Like a not-guilty verdict, failing to reject H₀ doesn't prove it's true; it means the evidence wasn't strong enough. That's why the claim you want to find evidence for always goes in Hₐ.
Writing hypotheses
The test is a one-sample z-test for a population proportion. H₀: p = p₀, where p₀ is the claimed value. Hₐ is one of:
- Hₐ: p < p₀ (one-sided, you suspect the true value is lower)
- Hₐ: p > p₀ (one-sided, you suspect it's higher)
- Hₐ: p ≠ p₀ (two-sided, you suspect it's different in either direction)
Rules for good hypotheses
Hypotheses are about parameters (p), never statistics (p̂). You already know p̂; there's nothing to test about it.
Define p in context: "p = the true proportion of all the company's deliveries that arrive on time." Name the response variable and the population.
Choose Hₐ from the research question before looking at the data. Even if H₀ is written with ≤ or ≥, the test is carried out at the boundary value p₀.
Use a two-sided alternative when the question says "different" or "changed" without a direction.
Wording to alternative
Match the wording of the question to Hₐ using the table below. Choose the significance level α before you collect data, too. The usual choice is 0.05; use 0.01 when a false alarm would be costly (topic 3.8).
| The question asks whether… | Hₐ |
|---|---|
| the rate is lower than claimed | p < p₀ |
| more than half, a majority | p > 0.5 |
| the rate has increased | p > p₀ |
| the rate has changed or differs | p ≠ p₀ |
Conditions
The conditions match the interval's, with one difference: the normal check uses the null value p₀, not the observed counts np̂ and n(1 − p̂).
- Random: the data come from a random sample from the population (name it in context).
- 10%: when sampling without replacement, the sample is no more than 10% of the population (n ≤ 0.10N).
- Normal: np₀ ≥ 10 and n(1 − p₀) ≥ 10, using the null value, because the test assumes H₀ is true.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Set up a one-sided test
A delivery company claims 90% of its packages arrive on time. A consumer group suspects the rate is lower. It selects a random sample of 150 of the company's many thousands of deliveries last month. Identify the test, state the hypotheses and check the conditions.
Show the solutionHide the solution
- Step 1: Test: one-sample z-test for a population proportion.
- Step 2: Parameter: p = the true proportion of all the company's deliveries last month that arrived on time.
- Step 3: H₀: p = 0.90. Hₐ: p < 0.90 (the group suspects lower).
- Step 4: Random: random sample of 150 deliveries.
- Step 5: 10%: 150 is far less than 10% of the many thousands of deliveries.
- Step 6: Normal: np₀ = 150(0.90) = 135 ≥ 10 and n(1 − p₀) = 150(0.10) = 15 ≥ 10.
Answer: One-sample z-test for p, the true proportion of last month's deliveries that arrived on time. H₀: p = 0.90; Hₐ: p < 0.90. All three conditions are met.
- Example 2Calculator allowed
Trap: hypotheses about the statistic
In a random sample of 500 voters, 54% support a candidate. A student writes H₀: p̂ = 0.50, Hₐ: p̂ > 0.54. Fix the hypotheses for testing whether a majority of all voters support the candidate.
Show the solutionHide the solution
- Step 1: Hypotheses must be about the parameter p, not p̂.
- Step 2: The sample result (0.54) never appears in the hypotheses.
- Step 3: "Majority" means more than 0.50.
Answer: H₀: p = 0.50 and Hₐ: p > 0.50, where p is the true proportion of all voters who support the candidate.
Common mistakes
- Writing hypotheses with p̂ or with the sample value.
- Checking the normal condition with p̂ instead of p₀ for a test.
- Choosing the direction of Hₐ after looking at the data.
- Leaving the parameter undefined or defining it for the sample.
On the exam
- The inference question usually earns a point for correct hypotheses with a defined parameter. Use symbols and words: "p = the true proportion of all…"
- If a question asks whether the data give evidence that something has "changed," use a two-sided alternative.
Connected topics
Videos
Check yourself
3 questions on 3.5 Setting Up a Test for a Population Proportion. Pick an answer to see if you got it, and why.
A test of H₀: p = 0.08 versus Hₐ: p > 0.08 is planned with a random sample of 100 items from a large shipment. Why is the large counts condition not met?
A coin-flipping machine is supposed to produce heads 50% of the time. An engineer wants to check whether the machine is off in either direction. Which alternative hypothesis should she use?
A city's mayor claims that 60% of the city's voters approve of the job she is doing. A reporter suspects the true percent is lower. In a random sample of 400 city voters, 222 (55.5%) say they approve.
Described scenario with invented results
Which are the correct hypotheses?
0 of 3 answered