Setting Up Tests for Population Proportion - AP Statistics

A streaming service reports that the proportion of subscribers who use the family plan is $0.30$ . A consumer group wonders if the true proportion is different. In a random sample of $n=500$ subscribers, $170$ use the family plan (sample proportion $\hat{p}=0.340$ ). Which hypotheses are appropriate?
Answer: $H_0: p=0.30$ ; $H_a: p\ne 0.30$ This question requires setting up a two-tailed test for a population proportion. The streaming service reports a specific value (0.30), and the consumer group wonders if the true proportion is "different," indicating a two-tailed test. The null hypothesis is H₀: p = 0.30, and the alternative is $H_a$ : p ≠ 0.30. Choice C correctly shows these hypotheses. The sample proportion of 0.340 might suggest the true value is higher, but since we're testing for any difference, we use the two-sided alternative with ≠. Never use sample statistics (p̂) in hypotheses or reverse the null and alternative (as in choice E). The word "different" without specifying a direction always indicates a two-tailed test.
A restaurant chain reports that 70% of customers rate their experience as "satisfied" or "very satisfied." A random sample of 150 customers is surveyed, and 96 give a satisfied rating ( $\hat{p}=0.64$ ). Which hypotheses are appropriate for determining whether the reported proportion is accurate?
Answer: $H_0: p=0.70\quad H_a: p\ne0.70$ The skill assessed is setting up hypotheses for testing a reported proportion of 0.70 in AP Statistics, to determine accuracy. The hypotheses H0: p = 0.70 and Ha: p ≠ 0.70 fit a two-sided test for verifying the exact value, as in choice A. A distractor is choice D, substituting ˆp for p. Choice C uses a one-sided alternative, unsuitable for accuracy checks. For a mini-lesson, use ≠ in Ha for assessing reported or exact values to allow rejection for any difference. Null includes equality to the reported figure, about p. This setup evaluates if the proportion matches without directional bias.
A school district states that the proportion of its high school students who participate in at least one extracurricular activity is no more than $0.60$ . A random sample of $n=200$ students finds $134$ who participate (sample proportion $\hat{p}=0.670$ ). Which hypotheses are appropriate for testing the district's statement?
Answer: $H_0: p=0.60$ ; $H_a: p>0.60$ This question tests hypothesis setup when the claim involves "no more than" language. The district states the proportion is "no more than 0.60," which means p ≤ 0.60. To test this claim, we place the boundary value with equality in the null hypothesis: H₀: p = 0.60. Since we're testing if the true proportion might be greater than claimed, the alternative is $H_a$ : p > 0.60. Choice A correctly shows this setup. The sample proportion of 0.670 supports the alternative, but this doesn't affect how we write the hypotheses. Common errors include using ≤ in the null hypothesis or using the sample proportion (0.67) instead of the claimed value (0.60). Remember: "no more than" claims lead to right-tailed tests.
A nonprofit organization states that 15% of households in a county have experienced food insecurity in the past year. A researcher thinks the true proportion is different from 15%. From a random sample of 400 households, 74 report food insecurity (so $ \hat{p}=74/400$). Which hypotheses are appropriate for a one-proportion $z$ test?
Answer: $H_0: p=0.15$ ; $H_a: p\ne 0.15$ This question evaluates hypotheses for thinking food insecurity differs from 15%, in a one-proportion z-test. Correct choice A: H0: p = 0.15 and Ha: p ≠ 0.15, for a two-sided test on 'different from'. Distractors use sample p-hat = 0.185 in H0, like option B, or p-hat symbols in D. Option C swaps H0 and Ha. Mini-lesson: Two-sided tests use ≠ in Ha to check for deviation in either direction. Always put equality in H0, about population p. Sample data is separate, used for test statistics.
An online retailer claims that at least 90% of its packages arrive on time. A random sample of 300 recent shipments shows 261 arrived on time (so $ \hat{p}=261/300$). A consumer advocate wants to test whether the true on-time rate is below the claim. Which hypotheses are appropriate?
Answer: $H_0: p=0.90$ ; $H_a: p<0.90$ This question involves setting up hypotheses to test if an on-time delivery rate is below the claimed 90%, for a one-proportion z-test. Choice C is correct: H0: p = 0.90 and Ha: p < 0.90, reflecting the advocate's goal to show it's lower. Distractors include using the sample p-hat = 0.87 in H0, as in option A, or using p-hat symbols, like option D. Option B incorrectly places the inequality in H0, which should always have equality. Mini-lesson: When a claim is 'at least' a value, test Ha as less than to challenge it, with H0 as equal to the boundary value. This creates a left-tailed test. Hypotheses are about p, not observed sample proportions.
A test of $H_0:p=0.08$ uses $n=80$ . Which condition fails?
Answer: $np_0\ge10$ For a one-proportion z test, both np0 and n(1-p0) must be at least 10. Here 80 times 0.08 = 6.4, so np0 >= 10 fails. The other side is 80 times 0.92 = 73.6, which passes. The tempting mistake is checking n(1-p0) or the sample size, but those conditions are satisfied; the expected successes are too few.
A student wants to test if a coin is biased towards landing on heads. The student flips the coin 40 times. Which of the following is a required condition to perform a valid one-sample z-test for the proportion of heads?
Answer: Assuming the coin is fair, the expected number of heads and tails must both be at least 10. The Large Counts condition for a significance test requires using the hypothesized proportion $p_0$ . To test if a coin is biased, the null hypothesis is that it is fair, so $H_0: p = 0.5$ . The expected counts are $40(0.5) = 20$ heads and $40(0.5) = 20$ tails. Since both are at least 10, the condition is met. Using observed counts is incorrect.
A school principal believes that more than 60% of students at the school support a new dress code. A random sample of 200 students is taken, and 134 say they support it ( $\hat{p}=0.67$ ). Which hypotheses are appropriate for testing the principal's belief with a one-proportion test?
Answer: $H_0: p = 0.60\quad H_a: p > 0.60$ This question tests hypothesis setup when the claim is that 'more than' a certain percentage holds true. The principal believes more than 60% support the dress code, so we're testing whether p > 0.60. The null hypothesis must contain equality, so H₀: p = 0.60, and the alternative hypothesis is $H_a$ : p > 0.60, making this a right-tailed test. Choice C incorrectly uses the sample proportion symbol (p̂) in the hypotheses. Choice D wrongly uses the observed sample proportion 0.67 in the null hypothesis. Choice A incorrectly includes inequality in the null hypothesis (p ≥ 0.60), which violates the principle that H₀ must specify a single value. Remember that the null hypothesis always represents the status quo or boundary condition with equality, while the alternative represents what we're trying to find evidence for.
To challenge a claim that support is at least 40%, which setup is correct?
Answer: $H_0:p=0.40$ , $H_a:p<0.40$ Since the claim is support is at least 40%, you're trying to find evidence it's actually less. Set the null as equality at 0.40 and the alternative as p < 0.40. The tempting opposite setup p > 0.40 would test whether support exceeds the claim, which strengthens it rather than challenging it.
In the American legal system, a defendant is presumed innocent until proven guilty. If a criminal trial is viewed as a hypothesis test where the claim to be tested is that the defendant is guilty, what would be the null and alternative hypotheses?
Answer: $H_0$ : The defendant is innocent. $H_a$ : The defendant is guilty. The null hypothesis ( $H_0$ ) is the default assumption, which is the presumption of innocence. The alternative hypothesis ( $H_a$ ) is the claim for which we require overwhelming evidence to accept, which is guilt. Therefore, $H_0$ is innocence and $H_a$ is guilt.
A nonprofit claims that more than 55% of donors prefer to give online rather than by mail. A random sample of 220 donors is selected, and 112 prefer online giving ( $\hat{p}=0.509$ ). Which hypotheses are appropriate for testing the nonprofit's claim about the population proportion $p$ ?
Answer: $H_0: p=0.55\quad H_a: p>0.55$ This question examines hypothesis setup for a one-proportion z-test in AP Statistics, claiming p > 0.55. The correct hypotheses are H0: p = 0.55 and Ha: p > 0.55, aligning with the one-sided 'more than' claim, which is choice B. A frequent distractor is choice C, using ˆp wrongly. Choice A reverses to less than, mismatching the claim. In a mini-lesson, match Ha to the claim: > for more than, with H0 as equality. Focus on p, not sample estimates. One-sided tests suit directional claims like this.
A voter advocacy group states that at least 55% of registered voters in a county support a proposed bond measure. A journalist suspects support is actually less than 55%. In a random sample of 180 registered voters, 90 support the measure ( $\hat{p}=0.50$ ). Which hypotheses are appropriate for testing the journalist's suspicion?
Answer: $H_0: p = 0.55\quad H_a: p < 0.55$ This question tests understanding of hypothesis setup when challenging a claim about 'at least' a certain percentage. The advocacy group states at least 55% support the measure, and the journalist suspects it's less. The null hypothesis uses the boundary value with equality: H₀: p = 0.55. The alternative hypothesis reflects the journalist's suspicion: $H_a$ : p < 0.55, creating a left-tailed test. Choice B incorrectly uses the sample proportion 0.50 in the null hypothesis. Choice C incorrectly includes inequality (≥) in the null hypothesis. Choice D uses sample notation (p̂) instead of population parameter notation. When someone claims 'at least' a certain value and we suspect it's less, we test H₀: p = (boundary value) against $H_a$ : p < (boundary value).
A manufacturer of a new smartphone claims that 95% of its phones are defect-free. A consumer advocacy group suspects this claim is inflated and that the true proportion of defect-free phones is lower. Let $p$ be the true proportion of defect-free phones. Which set of hypotheses is appropriate for the consumer group's test?
Answer: $H_0: p = 0.95$ vs. $H_a: p < 0.95$ The consumer group is investigating if the proportion is lower than claimed. The claim value of 0.95 serves as the null hypothesis. The alternative hypothesis reflects the group's suspicion. Thus, the hypotheses are $H_0: p = 0.95$ and $H_a: p < 0.95$ .
A state election office claims that the proportion of registered voters who support a proposed ballot measure is less than $50\%$ . A polling organization takes a random sample of $n=500$ registered voters and finds $270$ support the measure (so $\hat{p}=0.54$ ). Which hypotheses are appropriate for testing the state office's claim?
Answer: $H_0: p=0.50$ ; $H_a: p<0.50$ This question requires careful reading to set up the correct hypotheses. The state office claims support is LESS THAN 50%, which is what we want to test. The null hypothesis uses the boundary value: H₀: p = 0.50, and the alternative reflects the claim: Hₐ: p < 0.50. Choice B has the wrong direction. Choice C incorrectly uses the sample proportion (p̂). Choice D reverses the hypotheses. Choice E uses the sample proportion value instead of the claimed threshold. Note that even though our sample shows 54% support (which contradicts the claim), we still set up hypotheses based on the original claim being tested, not the sample results.
A wildlife agency reports that 25% of households in a region regularly put out bird feeders. A researcher suspects the true proportion is less than 25%. In a random sample of 160 households, 32 report regularly putting out bird feeders ( $\hat{p}=0.20$ ). Which hypotheses are appropriate for testing the researcher's suspicion?
Answer: $H_0: p = 0.25\quad H_a: p < 0.25$ This question tests understanding of hypothesis setup when a researcher suspects the true proportion is less than a reported value. The wildlife agency reports 25%, and the researcher suspects it's less. The null hypothesis is H₀: p = 0.25, and the alternative hypothesis is $H_a$ : p < 0.25, creating a left-tailed test. Choice B incorrectly uses the sample proportion 0.20 in the null hypothesis. Choice C reverses the null and alternative hypotheses. Choice D uses sample notation (p̂) instead of population parameter notation. Choice E has the wrong direction for the alternative hypothesis. When testing a suspicion that something is 'less than' a reported value, always use < in the alternative hypothesis to reflect this directional claim.
A high school teacher wants to test the claim that 75% of students have a part-time job. The teacher surveys the first 60 students who enter the library on a Monday afternoon. What is the primary reason that a one-sample z-test for a proportion would be inappropriate for this study?
Answer: The sample is a convenience sample, not a random sample of the student population. The fundamental condition for inference is that the data must come from a random sample or randomized experiment to avoid bias. Surveying the first 60 students in the library is a convenience sample and is not representative of the entire student population. This sampling bias makes the test results invalid.
An environmental group claims that more than 55% of local voters support a ban on single-use plastic bags. A random sample of 300 registered voters finds that 156 support the ban (so $\hat{p}=0.52$ ). Which hypotheses are appropriate for testing the group's claim about the population proportion $p$ ?
Answer: $H_0: p=0.55$ ; $H_a: p>0.55$ This question tests setting up hypotheses for a claim about "more than" a certain percentage. The environmental group claims MORE THAN 55% support the ban, which requires a right-tailed test with H₀: p = 0.55 and Hₐ: p > 0.55. Option C incorrectly uses p̂ instead of p, option D incorrectly uses the observed sample proportion 0.52, and options A and E have the wrong direction. The fact that the sample proportion (0.52) is less than the claimed value (0.55) doesn't change how we set up the hypotheses - we're still testing the original claim. Always base your hypotheses on the claim being made, not on what the sample shows.
A political analyst claims that less than 20% of eligible voters in a county are registered as independent. A random sample of 300 registered voters is taken, and 72 are independent ( $\hat{p}=0.24$ ). Which hypotheses are appropriate for testing the analyst's claim about the population proportion $p$ ?
Answer: $H_0: p=0.20\quad H_a: p<0.20$ This problem focuses on hypothesis formulation in AP Statistics for a claim that p < 0.20. The appropriate hypotheses are H0: p = 0.20 and Ha: p < 0.20, matching the one-sided claim, found in choice B. A common distractor is choice C, using ˆp instead of p. Choice A reverses the direction, testing greater than instead. In a mini-lesson, set Ha to reflect the claim's inequality, like < for less than, with H0 as equality. Use p in statements, reserving ˆp for calculations. Note that sample results don't alter the setup, even if they suggest otherwise.
A manufacturer claims that exactly 5% of the light bulbs it produces are defective. A quality-control inspector thinks the defect rate has changed. In a random sample of 400 bulbs, 28 are defective ( $\hat{p}=0.07$ ). Which hypotheses are appropriate to test whether the defect rate has changed using a one-proportion test?
Answer: $H_0: p = 0.05\quad H_a: p \ne 0.05$ This question involves testing whether a proportion has changed from a specific value, requiring a two-tailed test. The manufacturer claims exactly 5% defect rate, so H₀: p = 0.05. Since the inspector thinks the rate 'has changed' (could be higher or lower), the alternative hypothesis is $H_a$ : p ≠ 0.05. Choice B incorrectly uses the sample proportion 0.07 in the null hypothesis. Choice C wrongly uses sample proportion notation (p̂) instead of the population parameter. Choice D reverses the null and alternative hypotheses. Choice E sets up a one-tailed test instead of the required two-tailed test. When the research question asks whether something 'has changed' or 'is different,' always use a two-tailed alternative hypothesis with the ≠ symbol.
A public health official claims that at most 10% of households in a county have elevated radon levels. A random sample of 180 households is tested, and 27 have elevated radon (so $\hat{p}=0.15$ ). Which hypotheses are appropriate for testing whether the true population proportion $p$ is higher than the official's claim?
Answer: $H_0: p=0.10$ ; $H_a: p>0.10$ This question tests setting up hypotheses for a claim about "at most" 10%. The official claims AT MOST 10% have elevated radon, and we want to test if the true proportion is HIGHER than this claim. This requires a right-tailed test with H₀: p = 0.10 and Hₐ: p > 0.10. Option C incorrectly uses the sample statistic p̂ in the hypotheses, option D incorrectly uses the observed sample proportion 0.15 as the hypothesized value, and option E has the wrong direction. When someone claims "at most" a certain percentage and you want to test if it's actually higher, use a right-tailed test. The phrase "at most 10%" becomes the null hypothesis value of 0.10.

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