AP Statistics Quiz: Difference Of Two Population Proportions Setup

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Difference Of Two Population Proportions SetupQuestion 1 of 20

A city compares two methods for repairing potholes to see which results in fewer repairs needed within 6 months. For Method A, 19 of 85 repaired potholes needed another repair within 6 months. For Method B, 10 of 80 repaired potholes needed another repair within 6 months. The city's research claim is that Method A has a higher proportion needing another repair than Method B. Which hypotheses are appropriate for testing this claim about the population proportions?

H_0: p_A-p_B=0\quad\text{vs.}\quad H_a: p_A-p_B<0

H_0: p_A-p_B=0\quad\text{vs.}\quad H_a: p_A-p_B>0

H_0: \hat{p}_A-\hat{p}_B=0\quad\text{vs.}\quad H_a: \hat{p}_A-\hat{p}_B>0

H_0: p_B-p_A=0\quad\text{vs.}\quad H_a: p_B-p_A>0

H_0: p_A=0.20\quad\text{vs.}\quad H_a: p_A>0.20

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AP Statistics Quiz

AP Statistics Quiz: Difference Of Two Population Proportions Setup

Practice Difference Of Two Population Proportions Setup in AP Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Difference Of Two Population Proportions Setup, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

$H_0: p_A-p_B=0\quad\text{vs.}\quad H_a: p_A-p_B<0$
$H_0: p_A-p_B=0\quad\text{vs.}\quad H_a: p_A-p_B>0$ (correct answer)
$H_0: \hat{p}_A-\hat{p}_B=0\quad\text{vs.}\quad H_a: \hat{p}_A-\hat{p}_B>0$
$H_0: p_B-p_A=0\quad\text{vs.}\quad H_a: p_B-p_A>0$
$H_0: p_A=0.20\quad\text{vs.}\quad H_a: p_A>0.20$

Explanation: The city claims Method A has a higher proportion needing repairs than Method B, which is a directional claim requiring a one-sided test. The alternative hypothesis should be p_A - p_B > 0. Option A incorrectly uses p_A - p_B < 0, which would mean Method A is better, contradicting the claim. Option C uses sample statistics (p̂) instead of population parameters. Option D reverses the order to p_B - p_A > 0, which is mathematically equivalent to p_A - p_B < 0, again contradicting the claim. Option E tests only Method A against a fixed value rather than comparing methods. When setting up hypotheses, carefully match the direction of the inequality to the research claim about which group has the higher proportion.

Question 2

An engineer tests whether a new manufacturing process changes the proportion of parts that are defective. Under the old process, 17 of 220 randomly selected parts are defective. Under the new process, 29 of 210 randomly selected parts are defective. The engineer's research claim is that the defect proportion is different under the new process than under the old process. Which hypotheses are appropriate for testing this claim?

$H_0: p_{new}-p_{old}=0\quad\text{vs.}\quad H_a: p_{new}-p_{old}>0$
$H_0: p_{new}-p_{old}=0\quad\text{vs.}\quad H_a: p_{new}-p_{old}\ne 0$ (correct answer)
$H_0: \hat{p}_{new}-\hat{p}_{old}=0\quad\text{vs.}\quad H_a: \hat{p}_{new}-\hat{p}_{old}\ne 0$
$H_0: p_{old}-p_{new}=0\quad\text{vs.}\quad H_a: p_{old}-p_{new}\ne 0$
$H_0: p_{new}=0.10\quad\text{vs.}\quad H_a: p_{new}\ne 0.10$

Explanation: The engineer claims the defect proportion is different under the new process, without specifying if it's higher or lower. This requires a two-sided alternative hypothesis: p_new - p_old ≠ 0. Option A incorrectly uses a one-sided test (p_new > p_old) when no direction is specified. Option C uses sample statistics (p̂) instead of population parameters. Option D reverses the order but maintains the two-sided test, which would also be mathematically acceptable. Option E tests only the new process against a fixed proportion rather than comparing processes. When testing if a change in process affects a proportion without specifying the direction of change, use a two-sided alternative to detect differences in either direction.

Question 3

A school nurse wants to know whether the proportion of students who report getting at least 8 hours of sleep on school nights differs between students who participate in after-school sports and students who do not. In a random sample, 78 of 150 sports participants reported at least 8 hours, and 64 of 160 non-participants reported at least 8 hours. The nurse's research claim is that the true proportions are different. Which hypotheses are appropriate?

$H_0: \hat{p}_{\text{sports}}-\hat{p}_{\text{no sports}}=0$ ; $H_a: \hat{p}_{\text{sports}}-\hat{p}_{\text{no sports}}\ne 0$
$H_0: p_{\text{sports}}=p_{\text{no sports}}$ ; $H_a: p_{\text{sports}}\ne p_{\text{no sports}}$
$H_0: p_{\text{sports}}-p_{\text{no sports}}=0$ ; $H_a: p_{\text{sports}}-p_{\text{no sports}}\ne 0$ (correct answer)
$H_0: p_{\text{no sports}}-p_{\text{sports}}=0$ ; $H_a: p_{\text{no sports}}-p_{\text{sports}}>0$
$H_0: p_{\text{sports}}=0.50$ ; $H_a: p_{\text{sports}}\ne 0.50$

Explanation: This question tests your ability to set up hypotheses for comparing two population proportions. The nurse wants to test if the proportion of students getting at least 8 hours of sleep differs between sports participants and non-participants, which requires a two-tailed test. The common mistake (option A) is using sample proportions ( $\hat{p}$ ) in the hypotheses instead of population proportions ( $p$ ). When writing hypotheses, we always use population parameters ( $p$ ), not sample statistics ( $\hat{p}$ ). The correct setup uses $p_1 - p_2 = 0$ for $H_0$ and $p_1 - p_2 \ne 0$ for $H_a$ , which is mathematically equivalent to option B but option C follows the standard format for difference tests.

Question 4

A company compares two website designs to see which leads to a higher purchase rate. Of 200 randomly selected visitors shown Design 1, 34 made a purchase. Of 180 randomly selected visitors shown Design 2, 18 made a purchase. The company's research claim is that Design 1 has a higher purchase proportion than Design 2. Which hypotheses are appropriate for a test of the company's claim about the population proportions?

$H_0: p_1-p_2=0\quad\text{vs.}\quad H_a: p_1-p_2>0$ (correct answer)
$H_0: \hat{p}_1-\hat{p}_2=0\quad\text{vs.}\quad H_a: \hat{p}_1-\hat{p}_2>0$
$H_0: p_2-p_1=0\quad\text{vs.}\quad H_a: p_2-p_1>0$
$H_0: p_1=0.17\quad\text{vs.}\quad H_a: p_1>0.17$
$H_0: p_1-p_2=0\quad\text{vs.}\quad H_a: p_1-p_2\ne 0$

Explanation: This problem requires setting up hypotheses to test if Design 1 has a higher purchase rate than Design 2. The research claim is directional (Design 1 > Design 2), so we need a one-sided alternative hypothesis with p_1 - p_2 > 0. Option B incorrectly uses sample statistics (p̂) instead of population parameters. Option C reverses the order, testing if Design 2 > Design 1, which contradicts the claim. Option D tests only one proportion against a fixed value rather than comparing two populations. Option E uses a two-sided alternative when the claim specifies a direction. Remember that hypothesis tests always use population parameters (p) not sample statistics (p̂), and the alternative hypothesis must match the direction of the research claim.

Question 5

A public health team is investigating whether the proportion of residents who regularly wear a seat belt differs between residents in rural counties and residents in urban counties. In a survey, 162 of 200 rural residents reported regularly wearing a seat belt, and 190 of 220 urban residents reported regularly wearing a seat belt. The research claim is that the proportions differ. Which hypotheses are appropriate?

$H_0: p_{\text{rural}}-p_{\text{urban}}=0$ ; $H_a: p_{\text{rural}}-p_{\text{urban}}\ne 0$ (correct answer)
$H_0: p_{\text{urban}}-p_{\text{rural}}=0$ ; $H_a: p_{\text{urban}}-p_{\text{rural}}<0$
$H_0: \hat{p}_{\text{rural}}=\hat{p}_{\text{urban}}$ ; $H_a: \hat{p}_{\text{rural}}\ne \hat{p}_{\text{urban}}$
$H_0: p_{\text{rural}}=p_{\text{urban}}$ ; $H_a: p_{\text{rural}}>p_{\text{urban}}$
$H_0: p_{\text{rural}}=0.80$ ; $H_a: p_{\text{rural}}\ne 0.80$

Explanation: The public health team wants to test if seat belt usage proportions differ between rural and urban residents, requiring a two-tailed test. The research claim is that proportions differ (not which is higher), so we use ≠ in the alternative hypothesis. Option B incorrectly uses a one-tailed test with a specific direction. Option C uses sample proportions (p̂) instead of population proportions. Option D also uses a one-tailed test when no direction is specified. When the research question asks if proportions 'differ' without specifying a direction, always use a two-tailed test with p₁ - p₂ ≠ 0 in the alternative hypothesis.

Question 6

A researcher is studying whether the proportion of adults who have received a flu shot this year is lower among adults who work from home than among adults who work primarily on-site. In a random sample, 55 of 130 work-from-home adults had a flu shot, and 78 of 150 on-site adults had a flu shot. The research claim is that the work-from-home proportion is lower. Which hypotheses are appropriate?

$H_0: p_{\text{WFH}}-p_{\text{on-site}}=0$ ; $H_a: p_{\text{WFH}}-p_{\text{on-site}}<0$ (correct answer)
$H_0: \hat{p}_{\text{WFH}}-\hat{p}_{\text{on-site}}=0$ ; $H_a: \hat{p}_{\text{WFH}}-\hat{p}_{\text{on-site}}<0$
$H_0: p_{\text{on-site}}-p_{\text{WFH}}=0$ ; $H_a: p_{\text{on-site}}-p_{\text{WFH}}<0$
$H_0: p_{\text{WFH}}=p_{\text{on-site}}$ ; $H_a: p_{\text{WFH}}\ne p_{\text{on-site}}$
$H_0: p_{\text{WFH}}=0.50$ ; $H_a: p_{\text{WFH}}<0.50$

Explanation: This problem tests understanding of left-tailed hypothesis tests for two proportions. The researcher claims that the work-from-home proportion is lower than the on-site proportion, which translates to p_WFH < p_on-site. Rewriting this as p_WFH - p_on-site < 0 gives us the correct alternative hypothesis. Option B incorrectly uses sample proportions (p̂) in the hypotheses. Option C reverses the order, testing if on-site is lower than WFH. Remember that hypotheses always involve population parameters, and the order matters in one-tailed tests: the group claimed to be smaller comes first when using '<' in the alternative.

Question 7

A retailer is comparing two email subject lines to see which leads to a higher proportion of customers making a purchase within 24 hours. Of 500 customers emailed subject line A, 62 made a purchase; of 480 customers emailed subject line B, 45 made a purchase. The research claim is that subject line A has a higher purchase proportion. Which hypotheses are appropriate?

$H_0: p_A-p_B=0$ ; $H_a: p_A-p_B>0$ (correct answer)
$H_0: p_B-p_A=0$ ; $H_a: p_B-p_A>0$
$H_0: p_A=p_B$ ; $H_a: p_A\ne p_B$
$H_0: \hat{p}_A-\hat{p}_B=0$ ; $H_a: \hat{p}_A-\hat{p}_B>0$
$H_0: p_A=0.10$ ; $H_a: p_A>0.10$

Explanation: This question tests setting up a one-tailed hypothesis for comparing email subject line effectiveness. The retailer claims subject line A has a higher purchase proportion than B, making this a right-tailed test with p_A - p_B > 0 in the alternative hypothesis. Option B reverses the order, which would test if B is higher than A. Option C uses a two-tailed test when the claim specifies a direction. Option D incorrectly uses sample proportions (p̂) in the hypotheses. Remember that when a claim states one group is 'higher' or 'greater,' use a one-tailed test with the supposedly larger group first in the subtraction.

Question 8

A teacher wants to know whether offering optional review sessions increases the proportion of students who pass the final exam. In one class section with review sessions, 52 of 80 students passed. In another section without review sessions, 61 of 100 students passed. The teacher's research claim is that the review sessions increase the pass rate. Which hypotheses are appropriate for testing the teacher's claim about the population proportions?

$H_0: p_R-p_N=0\quad\text{vs.}\quad H_a: p_R-p_N>0$ (correct answer)
$H_0: p_R-p_N=0\quad\text{vs.}\quad H_a: p_R-p_N\ne 0$
$H_0: \hat{p}_R-\hat{p}_N=0\quad\text{vs.}\quad H_a: \hat{p}_R-\hat{p}_N>0$
$H_0: p_N-p_R=0\quad\text{vs.}\quad H_a: p_N-p_R>0$
$H_0: p_R=0.65\quad\text{vs.}\quad H_a: p_R>0.65$

Explanation: The teacher claims that review sessions increase the pass rate, so we need a one-sided test with the review group (R) having a higher proportion than the no-review group (N). The alternative hypothesis should be p_R - p_N > 0. Option B incorrectly uses a two-sided alternative when the claim is directional. Option C uses sample statistics (p̂) instead of population parameters. Option D reverses the order, testing if no-review is better than review, which contradicts the claim. Option E tests only the review group against a fixed value rather than comparing two groups. When testing if one treatment increases a proportion compared to another, the alternative hypothesis should reflect that directional claim with the appropriate inequality.

Question 9

A public health researcher compares smoking rates in two cities. In a random sample of 160 adults from City X, 44 are current smokers. In a random sample of 140 adults from City Y, 54 are current smokers. The researcher's claim is that the proportion of smokers differs between the two cities. Which hypotheses are appropriate for testing this claim using a difference in two population proportions?

$H_0: p_X-p_Y=0\quad\text{vs.}\quad H_a: p_X-p_Y>0$
$H_0: p_X-p_Y=0\quad\text{vs.}\quad H_a: p_X-p_Y\ne 0$ (correct answer)
$H_0: \hat{p}_X-\hat{p}_Y=0\quad\text{vs.}\quad H_a: \hat{p}_X-\hat{p}_Y\ne 0$
$H_0: p_X=0.275\quad\text{vs.}\quad H_a: p_X\ne 0.275$
$H_0: p_Y-p_X=0\quad\text{vs.}\quad H_a: p_Y-p_X>0$

Explanation: The researcher claims that smoking rates differ between the two cities, which requires a two-sided alternative hypothesis. The correct setup tests if p_X - p_Y ≠ 0, allowing for either city to have a higher rate. Option A incorrectly uses a one-sided test (p_X > p_Y) when no direction is specified. Option C uses sample statistics (p̂) instead of population parameters. Option D tests only City X's proportion against a specific value, not comparing two populations. Option E reverses the order and uses a one-sided test. When a research claim states that proportions "differ" without specifying which is larger, always use a two-sided alternative hypothesis (≠) to test for any difference in either direction.

Question 10

A restaurant chain wants to know whether changing its menu layout affects the proportion of customers who order a dessert. At 40 randomly selected locations using the old menu, 92 of 400 customers ordered dessert ( $\hat{p}_O=0.23$ ). At 35 independently selected locations using the new menu, 126 of 420 customers ordered dessert ( $\hat{p}_N=0.30$ ). The research claim is that the new menu changes (not necessarily increases) the dessert-ordering rate. Which hypotheses are appropriate?

$H_0: p_N-p_O=0$ ; $H_a: p_N-p_O\neq 0$ (correct answer)
$H_0: p_N-p_O=0$ ; $H_a: p_N-p_O>0$
$H_0: \hat{p}_N-\hat{p}_O=0$ ; $H_a: \hat{p}_N-\hat{p}_O\neq 0$
$H_0: p_O-p_N=0$ ; $H_a: p_O-p_N>0$
$H_0: p_N=0.30$ ; $H_a: p_N\neq 0.30$

Explanation: This question assesses hypothesis setup for two population proportions in AP Statistics, emphasizing a change without direction. Null H0: p_N - p_O = 0, no difference in dessert rates. Alternative Ha: p_N - p_O ≠ 0, rates differ, per choice A. Distractor B assumes a one-sided increase not stated in the claim. Choice C uses samples incorrectly. Mini-lesson: For 'changes' claims implying any difference, use two-sided ≠; define p_new and p_old; null always =0; avoid directional alternatives unless specified; hypotheses concern populations, not observed samples.

Question 11

A city is evaluating whether installing LED streetlights reduces the proportion of nighttime traffic accidents. In a random sample of 180 intersections with LED lights, 27 had a nighttime accident in the past year ( $\hat{p}_L=0.15$ ). In an independent random sample of 200 intersections with traditional lights, 44 had a nighttime accident ( $\hat{p}_T=0.22$ ). The research claim is that LED lights reduce the accident rate. Which hypotheses are appropriate?

$H_0: p_L-p_T=0$ ; $H_a: p_L-p_T<0$ (correct answer)
$H_0: p_T-p_L=0$ ; $H_a: p_T-p_L<0$
$H_0: p_L-p_T=0.07$ ; $H_a: p_L-p_T<0.07$
$H_0: \hat{p}_L-\hat{p}_T=0$ ; $H_a: \hat{p}_L-\hat{p}_T<0$
$H_0: p_L=0.15$ ; $H_a: p_L<0.15$

Explanation: This question evaluates setting up hypotheses for proportion differences in AP Statistics, with a reduction claim. Null: H0: p_L - p_T = 0, no difference in accident rates. Alternative: Ha: p_L - p_T < 0, LED lower, matching choice A. Distractor B reverses order, making direction wrong. Choice D uses samples. Mini-lesson: Set p1 for treatment (LED), p2 for traditional; null =0; < for reductions; parameters are population; verify order matches claim direction.

Question 12

A fitness app company claims that its new onboarding tutorial increases the proportion of new users who are still active after 30 days. In a random sample of 200 users who saw the tutorial, 118 were active after 30 days ( $\hat{p}_T=0.59$ ). In an independent random sample of 180 users who did not see the tutorial, 90 were active ( $\hat{p}_C=0.50$ ). The research claim is that the tutorial increases the 30-day active rate. Which hypotheses are appropriate?

$H_0: p_T-p_C=0$ ; $H_a: p_T-p_C>0$ (correct answer)
$H_0: p_T-p_C>0$ ; $H_a: p_T-p_C=0$
$H_0: \hat{p}_T-\hat{p}_C=0$ ; $H_a: \hat{p}_T-\hat{p}_C>0$
$H_0: p_C-p_T=0$ ; $H_a: p_C-p_T>0$
$H_0: p_T=0.59$ ; $H_a: p_T>0.59$

Explanation: This question evaluates the ability to formulate hypotheses for testing the difference between two population proportions in AP Statistics. The null hypothesis posits no difference in active rates between users with and without the tutorial, H0: p_T - p_C = 0. The alternative supports the claim that the tutorial increases the rate, so Ha: p_T - p_C > 0, corresponding to choice A. Distractors include choice C, which uses sample proportions instead of population ones, and choice D, which reverses the subtraction order and thus the direction of the claim. Choice B swaps null and alternative, which is invalid as the null must assume no effect. Mini-lesson: In two-proportion z-tests, identify the treatment and control groups, set null as p_treatment - p_control = 0, and choose one-sided alternative (>) for increase claims, ensuring parameters are population proportions, not samples.

Question 13

A university wants to test whether offering an optional practice exam increases the proportion of students who earn a B or higher in Calculus I. In a random sample of 110 students who took the practice exam, 72 earned a B or higher ( $\hat{p}_P\approx 0.655$ ). In an independent random sample of 130 students who did not take the practice exam, 70 earned a B or higher ( $\hat{p}_N\approx 0.538$ ). The research claim is that taking the practice exam increases the proportion earning a B or higher. Which hypotheses are appropriate?

$H_0: p_P-p_N=0$ ; $H_a: p_P-p_N>0$ (correct answer)
$H_0: p_P-p_N=0$ ; $H_a: p_P-p_N\neq 0$
$H_0: p_N-p_P=0$ ; $H_a: p_N-p_P>0$
$H_0: \hat{p}_P-\hat{p}_N=0$ ; $H_a: \hat{p}_P-\hat{p}_N>0$
$H_0: p_P=0.655$ ; $H_a: p_P>0.655$

Explanation: This question assesses setup for proportion differences in AP Statistics, increase claim. Null H0: p_P - p_N = 0, no difference in B+ rates. Alternative Ha: p_P - p_N > 0, practice higher, choice A. Distractor C reverses order. Choice D uses samples. Mini-lesson: p1 for treatment (practice); null =0; > for increases; populations; align with claim.

Question 14

A veterinarian claims that a lower proportion of dogs will have an upset stomach when fed Diet A compared with Diet B. In a sample of 150 dogs fed Diet A, 18 had an upset stomach; in a separate sample of 160 dogs fed Diet B, 30 had an upset stomach. Which hypotheses are appropriate for a test of this claim?

$H_0: p_A-p_B=0$ ; $H_a: p_A-p_B<0$ (correct answer)
$H_0: p_B-p_A=0$ ; $H_a: p_B-p_A<0$
$H_0: p_A-p_B=0$ ; $H_a: p_A-p_B>0$
$H_0: \hat p_A-\hat p_B=0$ ; $H_a: \hat p_A-\hat p_B<0$
$H_0: p_A=0.12$ ; $H_a: p_A<0.12$

Explanation: This problem involves testing if Diet A produces a lower proportion of upset stomachs than Diet B. The claim specifies 'lower,' requiring a less-than alternative hypothesis. The correct answer A uses H_a: p_A - p_B < 0, which tests if Diet A has a lower upset stomach rate. Choice D incorrectly uses sample proportions (p̂) in the hypotheses rather than population parameters. Choice C would test the opposite claim (Diet A causes more upset stomachs). Choice E incorrectly tests a single proportion against a specific value rather than comparing two diets. When setting up hypotheses for 'lower' claims, the group expected to be lower comes first in the subtraction, and the alternative uses the < symbol.

Question 15

A nonprofit is comparing two fundraising email subject lines to see if one leads to a higher donation rate. From a random sample of 500 recipients who received Subject Line 1, 65 donated ( $\hat{p}_1=0.13$ ). From an independent random sample of 480 recipients who received Subject Line 2, 48 donated ( $\hat{p}_2=0.10$ ). The research claim is that Subject Line 1 leads to a higher donation rate. Which hypotheses are appropriate?

$H_0: p_1-p_2=0$ ; $H_a: p_1-p_2>0$ (correct answer)
$H_0: p_1-p_2\neq 0$ ; $H_a: p_1-p_2=0$
$H_0: p_2-p_1=0$ ; $H_a: p_2-p_1>0$
$H_0: \hat{p}_1-\hat{p}_2=0$ ; $H_a: \hat{p}_1-\hat{p}_2>0$
$H_0: p_1=0.13$ ; $H_a: p_1>0.13$

Explanation: This question tests hypothesis formulation for two proportions in AP Statistics, claiming one higher. Null H0: p_1 - p_2 = 0, equal donation rates. Alternative Ha: p_1 - p_2 > 0, Line 1 higher, as in choice A. Distractor C reverses subtraction. Choice D uses samples. Mini-lesson: Define p1 for claimed better group; null =0; > for higher; populations only; avoid swapping null/alternative.

Question 16

A school district wants to know whether a new texting-based reminder system increases the proportion of parents who attend parent-teacher conferences. A random sample of 120 parents who received reminders had 78 attend ( $\hat{p}_R=0.65$ ), and an independent random sample of 150 parents who did not receive reminders had 75 attend ( $\hat{p}_N=0.50$ ). The district's research claim is that reminders increase the attendance rate. Which hypotheses are appropriate for a significance test comparing the two population proportions?

$H_0: \hat{p}_R-\hat{p}_N=0$ ; $H_a: \hat{p}_R-\hat{p}_N>0$
$H_0: p_R-p_N=0$ ; $H_a: p_R-p_N>0$ (correct answer)
$H_0: p_R-p_N=0.15$ ; $H_a: p_R-p_N\neq 0.15$
$H_0: p_N-p_R=0$ ; $H_a: p_N-p_R>0$
$H_0: p_R=0.65$ ; $H_a: p_R>0.65$

Explanation: This question assesses the skill of setting up hypotheses for a significance test comparing the difference of two population proportions in AP Statistics. The null hypothesis should state that there is no difference between the population proportions of parents attending conferences with and without reminders, expressed as H0: p_R - p_N = 0. The alternative hypothesis reflects the research claim that reminders increase attendance, so Ha: p_R - p_N > 0, which matches choice B. A common distractor is choice A, which incorrectly uses sample proportions (\hat{p}) instead of population proportions (p) in the hypotheses. Another distractor is choice D, which reverses the order of subtraction, leading to an incorrect direction for the alternative. In a mini-lesson on two-proportion setup: always define p1 and p2 clearly for each group, ensure the null assumes equality (difference of zero), and align the alternative with the claim's direction—here, one-sided greater than since the claim specifies an increase.

Question 17

A public health researcher is studying whether a new vaccine reduces the proportion of people who get a particular illness during flu season. In a random sample of 300 vaccinated individuals, 24 got the illness ( $\hat{p}_V=0.08$ ). In an independent random sample of 260 unvaccinated individuals, 39 got the illness ( $\hat{p}_U=0.15$ ). The research claim is that vaccination reduces the illness rate. Which hypotheses are appropriate?

$H_0: p_V=p_U$ ; $H_a: p_V\neq p_U$
$H_0: p_U-p_V=0$ ; $H_a: p_U-p_V<0$
$H_0: p_V-p_U=0$ ; $H_a: p_V-p_U<0$ (correct answer)
$H_0: \hat{p}_V-\hat{p}_U=0$ ; $H_a: \hat{p}_V-\hat{p}_U<0$
$H_0: p_V=0.08$ ; $H_a: p_V<0.08$

Explanation: This question tests hypothesis setup for the difference of two population proportions in AP Statistics, focusing on a reduction claim. The null is H0: p_V - p_U = 0, assuming no difference in illness rates between vaccinated and unvaccinated. The alternative Ha: p_V - p_U < 0 aligns with the claim that vaccination reduces the rate, as in choice C. A key distractor is choice B, which reverses the subtraction and direction, testing whether students correctly assign the order. Choice D uses sample hats incorrectly, while choice E focuses on one proportion alone. Mini-lesson: For two-proportion tests, define p1 for the treatment group (e.g., vaccinated) and p2 for control; null is always difference = 0; alternative uses < for reduction claims, and remember hypotheses are about populations, not samples.

Question 18

A political scientist suspects that the proportion of voters who support a ballot measure differs between urban and rural counties. A random sample of 250 urban voters found 140 in favor ( $\hat{p}_U=0.56$ ), and an independent random sample of 220 rural voters found 99 in favor ( $\hat{p}_R=0.45$ ). The research claim is that support is different between the two populations. Which hypotheses are appropriate?

$H_0: p_U-p_R=0$ ; $H_a: p_U-p_R\neq 0$ (correct answer)
$H_0: p_U-p_R=0$ ; $H_a: p_U-p_R>0$
$H_0: \hat{p}_U-\hat{p}_R=0$ ; $H_a: \hat{p}_U-\hat{p}_R\neq 0$
$H_0: p_R-p_U=0$ ; $H_a: p_R-p_U>0$
$H_0: p_U=0.56$ ; $H_a: p_U\neq 0.56$

Explanation: This question examines hypothesis formulation for comparing two population proportions in AP Statistics, with a non-directional difference claim. The null H0: p_U - p_R = 0 assumes equal support rates in urban and rural areas. The alternative Ha: p_U - p_R ≠ 0 reflects the claim of a difference without specifying direction, matching choice A. Distractor choice B adds a direction (>0) not supported by the claim, while choice C uses sample proportions. Choice D reverses order but keeps two-sided, potentially confusing subtraction. Mini-lesson: In two-proportion setups, choose two-sided alternative (≠) for 'differs' claims; define p1 and p2 consistently; null is always zero difference; avoid mixing with sample stats or adding unclaimed directions.

Question 19

A teacher is comparing two review methods to see which leads to a higher pass rate on a certification exam. In a random sample of 90 students using Method A, 63 passed ( $\hat{p}_A=0.70$ ). In an independent random sample of 100 students using Method B, 60 passed ( $\hat{p}_B=0.60$ ). The teacher's research claim is that Method A has a higher pass rate than Method B. Which hypotheses are appropriate?

$H_0: p_A-p_B=0$ ; $H_a: p_A-p_B>0$ (correct answer)
$H_0: p_A-p_B=0$ ; $H_a: p_A-p_B\neq 0$
$H_0: p_B-p_A=0$ ; $H_a: p_B-p_A>0$
$H_0: \hat{p}_A-\hat{p}_B=0$ ; $H_a: \hat{p}_A-\hat{p}_B>0$
$H_0: p_A=0.70$ ; $H_a: p_A>0.70$

Explanation: This question targets the skill of hypothesizing differences in population proportions for AP Statistics, with a directional higher claim. Null: H0: p_A - p_B = 0, no difference in pass rates. Alternative: Ha: p_A - p_B > 0, Method A higher, as in choice A. Distractor choice C reverses subtraction, flipping the direction incorrectly. Choice D uses sample proportions, a common error. Mini-lesson: Assign p1 to the group claimed superior (A here), p2 to the other; null difference = 0; one-sided > for 'higher' claims; ensure population parameters, not samples; two-sided only if no direction specified.

Question 20

A city transportation office claims that the proportion of commuters who are satisfied with bus service is higher among riders on routes with new buses than among riders on routes with older buses. In a sample, 92 of 140 riders on new-bus routes reported being satisfied, and 80 of 150 riders on old-bus routes reported being satisfied. Which hypotheses are appropriate?

$H_0: p_{\text{new}}-p_{\text{old}}=0$ ; $H_a: p_{\text{new}}-p_{\text{old}}>0$ (correct answer)
$H_0: \hat{p}_{\text{new}}-\hat{p}_{\text{old}}=0$ ; $H_a: \hat{p}_{\text{new}}-\hat{p}_{\text{old}}>0$
$H_0: p_{\text{old}}-p_{\text{new}}=0$ ; $H_a: p_{\text{old}}-p_{\text{new}}>0$
$H_0: p_{\text{new}}=p_{\text{old}}$ ; $H_a: p_{\text{new}}\ne p_{\text{old}}$
$H_0: p_{\text{new}}=0.60$ ; $H_a: p_{\text{new}}>0.60$

Explanation: This question requires setting up a one-tailed hypothesis test for comparing two population proportions. The transportation office claims that satisfaction is higher among new-bus riders than old-bus riders, making this a right-tailed test. The key is to match the order in the hypotheses to the claim: since we claim new > old, we write p_new - p_old > 0 in the alternative hypothesis. Option B incorrectly uses sample proportions (p̂) instead of population proportions. Option C reverses the order, which would test if old > new. When the claim involves 'higher' or 'greater than,' ensure your alternative hypothesis has the larger claimed group first.