Difference of Two Population Proportions (Setup) - AP Statistics

1. A sports scientist claims that a higher proportion of runners will experience muscle cramps when they drink water only compared with a sports drink during a race. In a random sample of 120 runners who drank water only, 42 reported cramps; in a separate random sample of 130 runners who drank a sports drink, 30 reported cramps. Which hypotheses are appropriate for a test of this claim?
   Answer: $H_0: p_{\text{water}}-p_{\text{sports}}=0$; $H_a: p_{\text{water}}-p_{\text{sports}}>0$ This question requires setting up hypotheses to test if water-only runners have a higher cramping rate than sports drink users. The claim specifies a direction (higher proportion), so we need a one-sided test. The correct answer A uses $H_a$: p_water - p_sports > 0, which tests if the water group has a higher cramping proportion. Choice D incorrectly uses sample proportions (p̂) in the hypotheses - we must use population parameters. Choice B reverses the subtraction order, which would test the opposite claim. Choice E incorrectly tests a single proportion against a fixed value rather than comparing two groups. When comparing two populations, the null hypothesis always states no difference (difference = 0), and the alternative reflects the research claim's direction.
2. 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?
   Answer: $H_0: p_1-p_2=0$; $H_a: p_1-p_2>0$ 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.
3. A hospital compares two treatments for preventing nausea after surgery. In a sample of 95 patients given Treatment 1, 21 experienced nausea. In a sample of 105 patients given Treatment 2, 34 experienced nausea. The hospital's research claim is that Treatment 1 results in a lower proportion of nausea than Treatment 2. Which hypotheses are appropriate for testing this claim about the population proportions?
   Answer: $H_0: p_1-p_2=0\quad\text{vs.}\quad H_a: p_1-p_2<0$ The hospital claims Treatment 1 results in lower nausea proportion than Treatment 2, meaning $p_1$ < $p_2$, which is equivalent to $p_1$ - $p_2$ < 0. Option B reverses the order but tests $p_2$ - $p_1$ < 0, which would mean $p_2$ < $p_1$ or Treatment 2 is better, contradicting the claim. Option C uses sample statistics (p̂) instead of population parameters. Option D tests only Treatment 1 against a fixed value rather than comparing treatments. Option E uses the wrong inequality direction (> instead of <). When one treatment is claimed to have a lower proportion of negative outcomes, the alternative hypothesis should reflect this with the appropriate inequality showing that treatment's proportion minus the other is less than zero.
4. 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?
   Answer: $H_0: p_R-p_N=0\quad\text{vs.}\quad H_a: p_R-p_N>0$ 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.
5. A restaurant owner claims that the proportion of customers who leave a tip is higher when the receipt includes a suggested tip amount than when it does not. One week, a random sample of 260 receipts with suggested tips showed 221 customers tipped; another week, a separate random sample of 240 receipts without suggested tips showed 190 customers tipped. Which hypotheses are appropriate for a test of this claim?
   Answer: $H_0: p_{\text{suggested}}-p_{\text{not}}=0$; $H_a: p_{\text{suggested}}-p_{\text{not}}>0$ This problem requires setting up hypotheses to test if suggested tip amounts increase tipping rates. The owner claims a 'higher' proportion tips with suggestions, indicating a one-sided test. The correct answer A uses $H_a$: p_suggested - p_not > 0, testing if the suggested tip group has a higher tipping proportion. Choice D incorrectly uses sample proportions (p̂) rather than population parameters. Choice B would test the opposite claim (suggested tips lead to lower tipping). Choice E incorrectly tests a single proportion against a fixed value instead of comparing two groups. In two-proportion tests, the null hypothesis always states the difference equals zero, while the alternative hypothesis direction (>, <, or ≠) depends on the research claim.
6. 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?
   Answer: $H_0: p_{new}-p_{old}=0\quad\text{vs.}\quad H_a: p_{new}-p_{old}\ne 0$ 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.
7. 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?
   Answer: $H_0: p_X-p_Y=0\quad\text{vs.}\quad H_a: p_X-p_Y\ne 0$ 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.
8. 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?
   Answer: $H_0: p_{\text{rural}}-p_{\text{urban}}=0$; $H_a: p_{\text{rural}}-p_{\text{urban}}\ne 0$ 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.
9. A marketing team claims that a lower proportion of customers will unsubscribe from a newsletter when the email subject line is personalized rather than generic. In a random sample of 500 recipients who got a personalized subject line, 55 unsubscribed; in a separate random sample of 480 recipients who got a generic subject line, 78 unsubscribed. Which hypotheses are appropriate for a test of this claim?
   Answer: $H_0: p_{\text{personalized}}-p_{\text{generic}}=0$; $H_a: p_{\text{personalized}}-p_{\text{generic}}<0$ This question tests hypothesis setup for a claim about a lower proportion of unsubscribes with personalized emails. The key is recognizing that 'lower proportion' means we need a less-than alternative hypothesis. The correct answer C uses $H_a$: p_personalized - p_generic < 0, which tests if the personalized group has a lower unsubscribe rate. Choice D incorrectly uses sample proportions (p̂) rather than population parameters. Choice B has the subtraction in the wrong order - it would test if generic has a lower rate than personalized. When testing for a 'lower' proportion, the group expected to be lower comes first in the subtraction, and the alternative hypothesis uses the < symbol. Always use population parameters (p) in hypotheses, never sample statistics (p̂).
10. 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?
    Answer: $H_0: p_A-p_B=0$; $H_a: p_A-p_B>0$ 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.
11. 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?
    Answer: $H_0: p_{\text{WFH}}-p_{\text{on-site}}=0$; $H_a: p_{\text{WFH}}-p_{\text{on-site}}<0$ 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.
12. 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?
    Answer: $H_0: p_N-p_O=0$; $H_a: p_N-p_O\neq 0$ 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.
13. 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?
    Answer: $H_0: p_T-p_C=0$; $H_a: p_T-p_C>0$ 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.
14. 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?
    Answer: $H_0: p_1-p_2=0\quad\text{vs.}\quad H_a: p_1-p_2>0$ 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.
15. 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?
    Answer: $H_0: p_U-p_R=0$; $H_a: p_U-p_R\neq 0$ 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.
16. A nutritionist claims that the proportion of clients who meet a daily fiber goal is higher for clients using a meal-planning app than for clients not using the app. In a sample, 68 of 120 app users met the goal, and 52 of 125 non-users met the goal. Which hypotheses are appropriate?
    Answer: $H_0: p_{\text{app}}-p_{\text{no app}}=0$; $H_a: p_{\text{app}}-p_{\text{no app}}>0$ This problem requires a right-tailed test since the nutritionist claims app users have a higher proportion meeting the fiber goal. The claim translates to p_app > p_no app, which we rewrite as p_app - p_no app > 0 for the alternative hypothesis. Option B reverses the order, testing if non-users are higher. Option D incorrectly uses sample proportions (p̂) rather than population proportions. Option E tests a single proportion against a fixed value, not a comparison between two groups. When setting up one-tailed tests, ensure the group claimed to be larger appears first in the subtraction for '>' alternatives.
17. 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?
    Answer: $H_0: p_A-p_B=0\quad\text{vs.}\quad H_a: p_A-p_B>0$ 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.
18. A political scientist compares support for a ballot measure between two age groups. In a random sample of 140 voters ages 18–34, 79 support the measure. In a random sample of 160 voters ages 35+, 78 support the measure. The research claim is that the proportion supporting the measure is higher among ages 18–34. Which hypotheses are appropriate for testing this claim using two population proportions?
    Answer: $H_0: p_{18-34}-p_{35+}=0\quad\text{vs.}\quad H_a: p_{18-34}-p_{35+}>0$ The research claim states that support is higher among ages 18-34, so we need a one-sided test with $p_{18-34}$ - $p_{35+}$ > 0. Option B incorrectly uses a two-sided alternative when the claim specifies direction. Option C uses sample statistics (p̂) instead of population parameters. Option D reverses the order to test if 35+ has higher support than 18-34, which contradicts the claim. Option E tests only the 18-34 group against a fixed proportion rather than comparing age groups. When comparing proportions between demographic groups with a directional claim, ensure the alternative hypothesis matches the stated direction and uses population parameters in the correct order.
19. A public health researcher wants to test the claim that a higher proportion of adults who received a text-message reminder got a flu shot than adults who did not receive a reminder. A random sample of 200 adults who received a reminder found 118 got a flu shot, and a separate random sample of 250 adults who did not receive a reminder found 120 got a flu shot. Which hypotheses are appropriate for a test of this claim?
    Answer: $H_0: p_{\text{reminder}}-p_{\text{no reminder}}=0$; $H_a: p_{\text{reminder}}-p_{\text{no reminder}}>0$ This question tests your understanding of setting up hypotheses for comparing two population proportions. The researcher wants to test if the proportion who got flu shots is higher among those who received reminders than those who didn't, making this a one-sided test. The correct answer B uses population parameters (p) rather than sample statistics (p̂), which is essential for hypothesis testing. Choice A incorrectly uses sample statistics (p̂) in the hypotheses - we always state hypotheses about population parameters, not sample statistics. The alternative hypothesis correctly shows p_reminder - p_no reminder > 0, which means the reminder group has a higher proportion. When comparing two proportions, always use population parameters (p) in hypotheses and ensure the order of subtraction matches the claim being tested.
20. 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?
    Answer: $H_0: p_P-p_N=0$; $H_a: p_P-p_N>0$ 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.

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