This AP Statistics question tests hypothesis formulation for two-sample mean differences, claiming caffeine reduces reaction time, so $\mu_{\text{caf}} < \mu_{\text{decaf}}$ or $H_a: \mu_{\text{caf}} - \mu_{\text{decaf}} < 0$. Choice A matches this with population parameters and correct order. Distractors include choice D, using sample means, which doesn't test populations. Choice E tests one group against its sample mean, missing the comparison. Mini-lesson: define $\mu_{\text{caf}}$ and $\mu_{\text{decaf}}$, null is $\mu_{\text{caf}} - \mu_{\text{decaf}} = 0$, alternative $< 0$ since caffeine is claimed to lower (faster) time compared to decaf.

This question tests setting up hypotheses for comparing mean LDL cholesterol between Mediterranean diet and usual diet groups. The nutritionist claims the Mediterranean diet "reduces" LDL, so we test if the Mediterranean group has lower mean LDL (μ_M - μ_U < 0). The null hypothesis states no difference (μ_M - μ_U = 0). Choice A uses a two-sided alternative when the claim is directional. Choice B reverses the inequality, testing if Mediterranean diet increases LDL. Choice C uses sample means (x̄) instead of population parameters. Choice E tests only the Mediterranean group against a fixed value rather than comparing two groups. For health outcomes where lower values are better (like cholesterol), "reduces" or "lowers" translates to testing if the treatment group mean is less than the control group mean.

This question tests setting up hypotheses for a two-sample t-test comparing mean vertical jump heights between plyometrics and usual training groups. The null hypothesis always states no difference between population means (μ_P - μ_U = 0), while the alternative reflects the research claim that plyometrics increases jump height, making it one-sided (μ_P - μ_U > 0). Choice A incorrectly uses sample means (x̄) instead of population parameters (μ) in the hypotheses. Choice C reverses the order of subtraction, which would test if usual training is better. Choice D tests only one group against a fixed value rather than comparing two groups. Choice E uses a two-sided alternative when the claim specifically states "increases," requiring a one-sided test. When setting up two-mean hypotheses, always use population parameters (μ), maintain consistent order of subtraction, and match the alternative hypothesis direction to the research claim.

This problem tests understanding of two-sided hypothesis tests when comparing plant heights under different lighting. The botanist claims the population mean heights "differ" without specifying which is taller, requiring a two-sided alternative (μ_LED - μ_Fl ≠ 0). The null hypothesis states no difference (μ_LED - μ_Fl = 0). Choice B incorrectly uses a one-sided alternative (>) when no direction is specified. Choice C uses sample means (x̄) instead of population parameters. Choice D reverses subtraction and uses one-sided test. Choice E tests only LED plants against a fixed value. When researchers claim two groups "differ" or are "different" without stating which is larger, always use a two-sided alternative hypothesis with the not-equal symbol.

This problem requires testing whether the express bus route decreases commute time compared to the regular route. The null hypothesis states no difference (μ_E - μ_R = 0), and since the claim is that express "decreases" time, we test if express has lower mean time (μ_E - μ_R < 0). Choice B incorrectly uses > which would test if express takes longer. Choice C reverses subtraction and uses the wrong inequality. Choice D uses sample means (x̄) instead of population parameters (μ). Choice E tests only the express route against a specific value. When testing if one treatment "decreases" or "reduces" a time-based outcome, the alternative hypothesis should reflect that the treatment group has a smaller population mean.

This question tests setting up hypotheses for the claim that yoga leads to lower resting heart rate than weight training. The claim is μ_Y < μ_W, which can be written as μ_Y - μ_W < 0. The null hypothesis assumes no difference: μ_Y - μ_W = 0. The alternative hypothesis supports the claim: μ_Y - μ_W < 0. Choice C correctly uses population parameters with the proper subtraction order to yield a < symbol. Choice D incorrectly uses sample statistics, and choice E incorrectly tests against a specific value. When testing if one treatment produces lower values, ensure the subtraction order in the alternative hypothesis produces the < symbol.

This question requires setting up a two-sided hypothesis test since the district claims the mean number of absences differs between start times without specifying which is higher. The null hypothesis states no difference: μ_8 - μ_9 = 0. The alternative hypothesis uses ≠ to indicate any difference: μ_8 - μ_9 ≠ 0. Choice B correctly uses population parameters with a two-sided alternative. Choice C incorrectly uses sample means instead of population means, and choice E incorrectly tests against a specific value. When a claim mentions a difference without specifying direction, always use a two-tailed test with ≠ in the alternative hypothesis.

This question evaluates hypotheses for two population means in AP Statistics, comparing blood pressure for low-sodium and no-change diets. The claim is low-sodium lowers mean BP (μ_L < μ_N), so H0: μ_L - μ_N = 0 and Ha: μ_L - μ_N < 0. Distractor choice B reverses order with <0, testing no-change lower, and choice D uses sample means. Choice E has >0, opposing the claim. Mini-lesson: Subscript clearly; null equality, <0 alternative for 'lower.' Ensure order matches inequality. Samples (126.3 vs. 131.7) guide tests, not statements.

This question probes setup for difference of means in AP Statistics, comparing plant heights under LED and fluorescent lights. The claim is LED leads to higher height (μ_LED > μ_F), so H0: μ_LED - μ_F = 0 and Ha: μ_LED - μ_F > 0. Distractor choice A reverses groups with <0, testing fluorescent higher, and choice C uses sample means. Choice E is two-sided, not matching directional claim. Mini-lesson: Use specific subscripts; null zero difference, >0 for 'higher.' Align subtraction to inequality. Sample means (34.5 vs. 33.8) inform but aren't in hypotheses.

This question involves testing whether caffeine leads to faster reactions (lower reaction times) compared to decaf. The null hypothesis states no difference (μ_caf - μ_decaf = 0), and since "faster reactions" means lower reaction time, the alternative is μ_caf - μ_decaf < 0. Choice B uses > which would test if caffeine makes reactions slower. Choice C reverses the subtraction order and uses the wrong inequality direction. Choice D incorrectly uses sample means (x̄) in the hypotheses. Choice E tests only the caffeinated group against a fixed value. For reaction time studies, "faster" or "quicker" responses correspond to lower numerical values, so improvement is tested with a less-than alternative hypothesis.