Slope of a Regression Model (Setup)
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AP Statistics › Slope of a Regression Model (Setup)
A conservationist studies whether the number of visitors to a park per day ($x$) predicts the amount of litter collected in pounds ($y$). The conservationist claims that more visitors leads to more litter in the population. Which hypotheses are appropriate for testing this claim about the slope in the regression of $y$ on $x$?
$H_0: \rho=0$ vs. $H_a: \rho>0$
$H_0: \beta=0$ vs. $H_a: \beta<0$
$H_0: b=0$ vs. $H_a: b>0$
$H_0: r=0$ vs. $H_a: r>0$
$H_0: \beta=0$ vs. $H_a: \beta>0$
Explanation
Testing setup for regression slope in AP Statistics. More visitors more litter, positive β, H₀: β = 0 vs Hₐ: β > 0, choice D. Distractors A and B correlations, C sample b, E negative. Lesson: β > 0 for increases. Match to claim. Assesses visitors' positive link to litter.
A social scientist studies whether the number of close friends a person reports ($x$) predicts a loneliness score ($y$), where higher scores mean more lonely. The scientist believes that more close friends is associated with lower loneliness in the population. Which hypotheses are appropriate for testing the claim about the slope in the regression of $y$ on $x$?
$H_0: \beta=0$ vs. $H_a: \beta<0$
$H_0: r=0$ vs. $H_a: r<0$
$H_0: b=0$ vs. $H_a: b<0$
$H_0: \rho=0$ vs. $H_a: \rho<0$
$H_0: \beta=0$ vs. $H_a: \beta>0$
Explanation
This question tests regression slope hypotheses in AP Statistics. More friends lowering loneliness score means negative slope, Ha: β < 0. B is correct: H0: β = 0 vs. Ha: β < 0. Distractors include b, r, ρ, or positive Ha. Mini-lesson: β < 0 for decreasing y with increasing x. Null is no relationship. Direction from association in claim.
A geologist studies whether the depth of an earthquake in km ($x$) predicts the amount of surface damage in dollars ($y$). The geologist believes that deeper earthquakes tend to cause less surface damage in the population. Which hypotheses are appropriate for testing this claim about the slope in the regression of $y$ on $x$?
$H_0: b=0$ vs. $H_a: b<0$
$H_0: \beta=0$ vs. $H_a: \beta>0$
$H_0: r=0$ vs. $H_a: r<0$
$H_0: \beta=0$ vs. $H_a: \beta<0$
$H_0: \rho=0$ vs. $H_a: \rho<0$
Explanation
Skill: hypothesizing about regression slope in AP Statistics. Deeper earthquakes less damage, negative β, H₀: β = 0 vs Hₐ: β < 0, choice A. Distractors B sample b, C and D correlations, E positive. Lesson: Negative slope for reduced y with x. Directional Hₐ. Evaluates depth's negative effect on damage.
A music teacher studies whether minutes of daily instrument practice ($x$) predicts performance rating ($y$) at a recital. The teacher believes that more practice improves performance ratings in the population. Which hypotheses are appropriate for testing the teacher’s claim about the slope of the population regression line predicting $y$ from $x$?
$H_0: \rho=0$ vs. $H_a: \rho>0$
$H_0: b=0$ vs. $H_a: b>0$
$H_0: \beta=0$ vs. $H_a: \beta>0$
$H_0: \beta=0$ vs. $H_a: \beta<0$
$H_0: r=0$ vs. $H_a: r>0$
Explanation
AP Statistics question on slope hypothesis. More practice improves ratings, positive β, H₀: β = 0 vs Hₐ: β > 0, choice B. Distractors A sample b, C and D correlations, E negative. Mini-lesson: Positive association uses > 0. Population focus. Tests practice's positive effect on performance.
A retailer studies whether the number of items displayed at the front of a store ($x$) predicts daily impulse purchases ($y$). The retailer believes that more items displayed increases impulse purchases in the population. Which hypotheses are appropriate for testing this claim about the population regression slope?
$H_0: b=0$ vs. $H_a: b>0$
$H_0: \rho=0$ vs. $H_a: \rho>0$
$H_0: \beta=0$ vs. $H_a: \beta<0$
$H_0: \beta=0$ vs. $H_a: \beta>0$
$H_0: r=0$ vs. $H_a: r>0$
Explanation
This question assesses the skill of setting up hypotheses for the slope of a regression model in AP Statistics, specifically for testing claims about the population regression slope. The retailer believes that more items displayed increases impulse purchases, indicating a positive association, so the alternative hypothesis should reflect a positive population slope β > 0, with the null being β = 0, which matches choice B. A common distractor is choice C, which incorrectly uses the sample slope b instead of the population parameter β, as hypotheses must focus on population parameters. Another distractor is choice A, which tests the sample correlation r, but the question is about the regression slope, not correlation. In a mini-lesson on slope hypothesis setup: The population regression model is y = α + βx + ε, where β represents the true slope; we test H0: β = 0 (no linear relationship) against Ha: β > 0 for positive associations or β < 0 for negative ones, ensuring the direction aligns with the research claim. Remember, ρ and r are for correlation tests, which are related but distinct from slope tests, as correlation measures strength and direction without quantifying the change per unit. Always verify that the hypotheses pertain to the population, not the sample.
A teacher investigates whether time spent on a new online practice system is related to students’ quiz scores. From a random sample of 35 students, the teacher records minutes spent practicing (explanatory variable $x$) and quiz score out of 20 (response variable $y$) and fits a least-squares regression line predicting score from minutes. The teacher’s claim is that there is some linear relationship (slope not equal to 0). Which hypotheses are appropriate for testing this claim about the population slope?
$H_0: \beta_1 \ne 0$ vs. $H_a: \beta_1 = 0$
$H_0: b_1 = 0$ vs. $H_a: b_1 \ne 0$
$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 \ne 0$
$H_0: r = 0$ vs. $H_a: r \ne 0$
$H_0: \beta_0 = 0$ vs. $H_a: \beta_0 \ne 0$
Explanation
This question evaluates the skill of formulating hypotheses for the population slope in a regression model in AP Statistics. The teacher's claim is that there is some linear relationship between practice time and quiz scores, without specifying direction, so the hypotheses are H₀: β₁ = 0 versus Hₐ: β₁ ≠ 0, indicating a two-sided test for any non-zero slope. This setup matches the non-directional claim of 'some' relationship. A common distractor is choice B, which uses correlation r instead of slope β₁, though r=0 is related but not the direct parameter for slope tests. Choice E incorrectly swaps null and alternative, which doesn't make sense for testing. Mini-lesson: hypotheses for regression slopes center on β₁, with null β₁=0; choose two-sided alternative for existence of relationship, one-sided for direction; avoid sample b₁ (choice C) or intercept β₀ (choice D), as they don't test population parameters.
A marine biologist studies whether ocean depth in meters ($x$) predicts water temperature in °C ($y$) at sampling locations. The biologist expects that greater depth corresponds to lower temperature in the population. Which hypotheses are appropriate for testing this expectation about the slope in the regression of $y$ on $x$?
$H_0: r=0$ vs. $H_a: r<0$
$H_0: b=0$ vs. $H_a: b<0$
$H_0: \rho=0$ vs. $H_a: \rho<0$
$H_0: \beta=0$ vs. $H_a: \beta<0$
$H_0: \beta=0$ vs. $H_a: \beta>0$
Explanation
This AP Statistics question focuses on hypothesizing about the regression slope. The biologist expects greater depth to lower temperature, implying negative β, so H₀: β = 0 versus Hₐ: β < 0 fits, per choice A. Distractors B, C, D use sample b or correlations r and ρ, and E has the opposite direction. Mini-lesson: β represents population slope; test H₀: β = 0 against directional Hₐ based on expectation. Greek symbols denote parameters. This evaluates if depth negatively affects temperature in oceanic populations.
A psychologist examines whether the number of hours of sleep a student gets is related to their reaction time on a computer task. For a random sample of 32 students, the psychologist records sleep hours (explanatory variable $x$) and reaction time in milliseconds (response variable $y$) and fits a regression predicting reaction time from sleep. The claim is that more sleep leads to faster reactions (smaller times), implying a negative population slope. Which hypotheses are appropriate for testing this claim about the slope?
$H_0: \beta_0 = 0$ vs. $H_a: \beta_0 < 0$
$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 \ne 0$
$H_0: r = 0$ vs. $H_a: r < 0$
$H_0: b_1 = 0$ vs. $H_a: b_1 < 0$
$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 < 0$
Explanation
This question targets the AP Statistics skill of hypothesis formulation for regression slopes. The psychologist claims more sleep leads to faster reaction times (lower ms), implying a negative slope, so H₀: β₁ = 0 versus Hₐ: β₁ < 0 is appropriate. This one-sided test matches the directional claim. Distractors include choice B with correlation r, which is related but not the slope parameter, and choice C using sample b₁. Choice D's two-sided test ignores the direction. Mini-lesson: set null β₁=0; alternative <0 for negative associations; use population β₁, not b₁, r, or β₀ (choice E), ensuring the test reflects the claim's direction.
A fitness app company studies whether the number of push-ups a user can do in one minute ($x$) predicts the user’s body fat percentage ($y$). The company claims that higher push-up counts are associated with lower body fat percentage in the population. Which hypotheses are appropriate for testing this claim about the slope of the population regression line predicting $y$ from $x$?
$H_0: r=0$ vs. $H_a: r<0$
$H_0: b=0$ vs. $H_a: b<0$
$H_0: \beta=0$ vs. $H_a: \beta<0$
$H_0: \rho=0$ vs. $H_a: \rho<0$
$H_0: \beta=0$ vs. $H_a: \beta>0$
Explanation
AP Statistics: setup for slope hypotheses. Higher push-ups lower body fat, negative β, H₀: β = 0 vs Hₐ: β < 0, choice D. Distractors A correlation ρ, B correlation r, C sample b, E positive. Mini-lesson: β < 0 when y decreases as x increases. Use population β. Tests push-ups' negative association with fat.
A veterinarian studies whether a dog’s age in years ($x$) predicts its activity level score ($y$), where higher scores mean more active. The veterinarian believes that older dogs tend to be less active in the population. Which hypotheses are appropriate for testing this claim about the slope of the population regression line predicting $y$ from $x$?
$H_0: \rho=0$ vs. $H_a: \rho<0$
$H_0: \beta=0$ vs. $H_a: \beta>0$
$H_0: \beta=0$ vs. $H_a: \beta<0$
$H_0: r=0$ vs. $H_a: r<0$
$H_0: b=0$ vs. $H_a: b<0$
Explanation
This question tests formulation of regression slope hypotheses in AP Statistics. Older dogs being less active means negative β, Ha: β < 0. A is accurate: H0: β = 0 vs. Ha: β < 0. Distractors use b, r, ρ, or positive Ha. Lesson: Negative association requires Ha < 0. β is population parameter. Infers from sample to population.