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.

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.

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.

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.

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.

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.

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.

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.

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.

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.