AP Statistics Quiz: Slope Of A Regression Model Setup

20 questions · exam conditions

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Slope Of A Regression Model SetupQuestion 1 of 20

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: r=0

vs.

H_a: r>0

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

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AP Statistics Quiz

AP Statistics Quiz: Slope Of A Regression Model Setup

Practice Slope Of A Regression Model 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 Slope Of A Regression Model Setup, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

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.

All questions

Question 1

$H_0: r=0$ vs. $H_a: r>0$
$H_0: \rho=0$ vs. $H_a: \rho>0$
$H_0: b=0$ vs. $H_a: b>0$
$H_0: \beta=0$ vs. $H_a: \beta>0$ (correct answer)
$H_0: \beta=0$ 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.

Question 2

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 $x$ ?

$H_0: b=0$ vs. $H_a: b<0$
$H_0: \beta=0$ vs. $H_a: \beta<0$ (correct answer)
$H_0: r=0$ vs. $H_a: r<0$
$H_0: \beta=0$ vs. $H_a: \beta>0$
$H_0: \rho=0$ vs. $H_a: \rho<0$

Explanation: 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.

Question 3

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 $x$ ?

$H_0: \beta=0$ vs. $H_a: \beta<0$ (correct answer)
$H_0: b=0$ vs. $H_a: b<0$
$H_0: r=0$ vs. $H_a: r<0$
$H_0: \rho=0$ vs. $H_a: \rho<0$
$H_0: \beta=0$ vs. $H_a: \beta>0$

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.

Question 4

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 $x$ ?

$H_0: b=0$ vs. $H_a: b>0$
$H_0: \beta=0$ vs. $H_a: \beta>0$ (correct answer)
$H_0: r=0$ vs. $H_a: r>0$
$H_0: \rho=0$ vs. $H_a: \rho>0$
$H_0: \beta=0$ vs. $H_a: \beta<0$

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.

Question 5

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?

$H_0: r=0$ vs. $H_a: r>0$
$H_0: \beta=0$ vs. $H_a: \beta>0$ (correct answer)
$H_0: b=0$ 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 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.

Question 6

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?

$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 \ne 0$ (correct answer)
$H_0: r = 0$ vs. $H_a: r \ne 0$
$H_0: b_1 = 0$ vs. $H_a: b_1 \ne 0$
$H_0: \beta_0 = 0$ vs. $H_a: \beta_0 \ne 0$
$H_0: \beta_1 \ne 0$ vs. $H_a: \beta_1 = 0$

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.

Question 7

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 $x$ ?

$H_0: \beta=0$ vs. $H_a: \beta<0$ (correct answer)
$H_0: b=0$ vs. $H_a: b<0$
$H_0: r=0$ vs. $H_a: r<0$
$H_0: \rho=0$ vs. $H_a: \rho<0$
$H_0: \beta=0$ 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.

Question 8

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?

$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 < 0$ (correct answer)
$H_0: r = 0$ vs. $H_a: r < 0$
$H_0: b_1 = 0$ vs. $H_a: b_1 < 0$
$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 \ne 0$
$H_0: \beta_0 = 0$ vs. $H_a: \beta_0 < 0$

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.

Question 9

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 $x$ ?

$H_0: \rho=0$ vs. $H_a: \rho<0$
$H_0: r=0$ vs. $H_a: r<0$
$H_0: b=0$ vs. $H_a: b<0$
$H_0: \beta=0$ vs. $H_a: \beta<0$ (correct answer)
$H_0: \beta=0$ 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.

Question 10

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 $x$ ?

$H_0: \beta=0$ vs. $H_a: \beta<0$ (correct answer)
$H_0: b=0$ vs. $H_a: b<0$
$H_0: r=0$ vs. $H_a: r<0$
$H_0: \rho=0$ vs. $H_a: \rho<0$
$H_0: \beta=0$ vs. $H_a: \beta>0$

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.

Question 11

A school administrator investigates whether class size is related to average end-of-year math score. For a random sample of 55 classes, the administrator records class size (explanatory variable $x$ ) and average math score (response variable $y$ ) and fits a regression line predicting score from class size. The claim is that larger classes are associated with lower scores, implying a negative population slope. Which hypotheses are appropriate for testing this claim about the slope of the population regression line?

$H_0: r = 0$ vs. $H_a: r < 0$
$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 < 0$ (correct answer)
$H_0: b_1 = 0$ vs. $H_a: b_1 < 0$
$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 > 0$
$H_0: \beta_0 = 0$ vs. $H_a: \beta_0 < 0$

Explanation: This question tests the skill of creating hypotheses for the population regression slope in AP Statistics. The administrator's claim is that larger classes associate with lower scores, implying a negative slope, so H₀: β₁ = 0 versus Hₐ: β₁ < 0 is correct. This one-sided alternative fits the directional claim. Distractors include choice A using r instead of β₁, and choice C with sample b₁. Choice D has the wrong direction (>0). Mini-lesson: null β₁=0 means no relationship; alternative <0 for negative claims; use β₁, not r, b₁, or β₀ (choice E), to properly test population associations.

Question 12

A sports scientist studies whether athletes' vertical jump height is related to their 40-yard dash time. For 25 randomly selected athletes, the scientist records vertical jump in inches (explanatory variable $x$ ) and dash time in seconds (response variable $y$ ) and fits a regression line predicting dash time from jump height. The claim is that greater jump height corresponds to faster times (smaller seconds), so the population slope should be negative. Which hypotheses are appropriate for testing this claim about the population regression slope?

$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 < 0$ (correct answer)
$H_0: r = 0$ vs. $H_a: r < 0$
$H_0: b_1 = 0$ vs. $H_a: b_1 < 0$
$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 \ne 0$
$H_0: \beta_0 = 0$ vs. $H_a: \beta_0 < 0$

Explanation: This question tests hypothesis setup for the regression slope in AP Statistics. The scientist claims that higher jump height corresponds to faster dash times (lower seconds), indicating a negative slope, so hypotheses are H₀: β₁ = 0 versus Hₐ: β₁ < 0. This matches the directional claim of negative association. A key distractor is choice B, using correlation r, which tests association but not specifically the slope parameter. Choice C uses sample b₁ incorrectly for population inference. Mini-lesson: for slope tests, null is β₁=0; alternative reflects claim direction (<0 here); distinguish from intercept β₀ (choice E) and ensure population parameter β₁, not b₁ or r; choice D's two-sided test doesn't fit the one-sided claim.

Question 13

A public health researcher studies whether neighborhoods with more walkable streets tend to have lower average body mass index (BMI). For a random sample of 60 neighborhoods, the researcher records walkability score (explanatory variable $x$ ) and mean adult BMI (response variable $y$ ) and fits the least-squares regression line to predict BMI from walkability. The research claim is that higher walkability is associated with lower mean BMI, meaning the population slope is negative. Which hypotheses are appropriate for testing this claim about the slope of the population regression line?

$H_0: r = 0$ vs. $H_a: r < 0$
$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 < 0$ (correct answer)
$H_0: b_1 = 0$ vs. $H_a: b_1 < 0$
$H_0: \beta_0 = 0$ vs. $H_a: \beta_0 < 0$
$H_0: \beta_1 = 0$ vs. $H_a: \beta_1 \ne 0$

Explanation: This question tests the skill of setting up hypotheses for the slope of a regression model in AP Statistics. The researcher's claim is that higher walkability is associated with lower mean BMI, implying a negative population slope, so the appropriate hypotheses are H₀: β₁ = 0 versus Hₐ: β₁ < 0, where β₁ is the true population slope. This aligns with a one-sided alternative hypothesis because the claim specifies a direction (negative slope). A common distractor here is choice A, which uses the correlation coefficient r instead of the slope β₁, but hypotheses for linear regression slopes focus on β₁, not r. Another distractor is choice C, which uses the sample slope b₁ instead of the population parameter β₁. In a mini-lesson on slope hypothesis setup: the null hypothesis typically states no linear relationship (β₁ = 0), while the alternative reflects the claimed direction or existence of a relationship; always use population parameters like β₁, not sample statistics like b₁ or r, and avoid confusing it with the intercept β₀ as in choice D.

Question 14

A marketing manager samples $n=50$ weeks and fits a least-squares regression line to predict weekly sales ( $y$ ) from advertising spending ( $x$ ). The manager wants to test whether advertising spending is linearly related to sales in the population, without specifying a direction. Which hypotheses are appropriate for testing this question about the population slope?

$H_0: r=0$ vs. $H_a: r\ne 0$
$H_0: b=0$ vs. $H_a: b\ne 0$
$H_0: \beta=0$ vs. $H_a: \beta\ne 0$ (correct answer)
$H_0: \beta=0$ vs. $H_a: \beta>0$
$H_0: \beta\ne 0$ vs. $H_a: \beta=0$

Explanation: The manager wants to test whether advertising is "linearly related" to sales without specifying direction, requiring a two-sided test. This non-directional question needs H₀: β = 0 versus Hₐ: β ≠ 0. Choice C correctly uses the population slope parameter β with a two-sided alternative. Choice B incorrectly uses sample slope b, and choice A uses correlation r. Choice D would be one-sided (positive only) and doesn't match the non-directional research question. When no direction is specified and researchers simply ask about "relationship" or "association," use a two-sided test with ≠ in the alternative hypothesis.

Question 15

A psychologist samples $n=27$ adults and fits a regression model predicting stress score ( $y$ ) from hours of sleep per night ( $x$ ). The psychologist claims that, in the population, more sleep is associated with lower stress (negative slope). Which hypotheses are appropriate for testing this claim about the population regression slope?

$H_0: \beta=0$ vs. $H_a: \beta<0$ (correct answer)
$H_0: r=0$ vs. $H_a: r<0$
$H_0: b=0$ vs. $H_a: b<0$
$H_0: \beta=0$ vs. $H_a: \beta\ne 0$
$H_0: \beta\le 0$ vs. $H_a: \beta>0$

Explanation: The psychologist claims more sleep is associated with lower stress, indicating a negative slope relationship. This directional claim requires H₀: β = 0 versus Hₐ: β < 0. Choice A correctly uses the population parameter β with the appropriate negative alternative hypothesis. Choice C incorrectly uses sample slope b instead of the population parameter. Choice B uses correlation r rather than slope β. Remember that hypothesis tests always concern population parameters (Greek letters), not sample statistics. When x increases and y decreases according to the claim, the alternative hypothesis should indicate β < 0.

Question 16

A real estate analyst records $n=35$ recent home sales and fits a least-squares regression line to predict sale price ( $y$ ) from square footage ( $x$ ). The analyst wants to test whether there is any linear relationship in the population (the slope could be positive or negative). Which hypotheses are appropriate for testing this question about the population slope?

$H_0: b=0$ vs. $H_a: b\ne 0$
$H_0: r=0$ vs. $H_a: r\ne 0$
$H_0: \beta=0$ vs. $H_a: \beta\ne 0$ (correct answer)
$H_0: \beta=0$ vs. $H_a: \beta>0$
$H_0: \beta\ne 0$ vs. $H_a: \beta=0$

Explanation: The analyst wants to test whether there is "any linear relationship" without specifying direction, which requires a two-sided test. This means testing whether the population slope β differs from zero in either direction. Choice C correctly sets up H₀: β = 0 versus Hₐ: β ≠ 0, using the population parameter. Choice A incorrectly uses sample slope b, and choice B uses correlation r. Choice D would be one-sided and doesn't match the non-directional research question. When researchers ask about "any relationship" or "whether related" without specifying increase/decrease, use a two-sided alternative hypothesis with ≠.

Question 17

An environmental engineer samples $n=40$ days and fits a least-squares regression line predicting daily electricity use ( $y$ , kWh) from daily high temperature ( $x$ , °F). The engineer claims that, in the population, hotter days lead to higher electricity use (positive slope). Which hypotheses are appropriate for testing this claim about the population slope?

$H_0: \beta=0$ vs. $H_a: \beta>0$ (correct answer)
$H_0: b=0$ vs. $H_a: b>0$
$H_0: r=0$ vs. $H_a: r>0$
$H_0: \beta\ge 0$ vs. $H_a: \beta<0$
$H_0: \beta=1$ vs. $H_a: \beta>1$

Explanation: The engineer claims hotter days lead to higher electricity use, indicating a positive slope relationship. This requires testing H₀: β = 0 (no relationship) against Hₐ: β > 0 (positive relationship). Choice A correctly uses the population parameter β with the appropriate one-sided positive alternative. Choice B incorrectly uses sample slope b instead of population parameter β. Choice C uses correlation r rather than slope. When testing claims about "higher," "increased," or "more" outcomes with increasing x-values, the alternative hypothesis should show β > 0. Always use Greek letters for population parameters in hypothesis statements.

Question 18

A botanist measures $n=25$ plants of the same species and uses linear regression to predict plant height ( $y$ , cm) from hours of sunlight per day ( $x$ ). The botanist claims that, in the population, additional sunlight increases average height, so the population regression slope is positive. Which hypotheses are appropriate for testing this claim about the slope?

$H_0: r=0$ vs. $H_a: r>0$
$H_0: \beta=0$ vs. $H_a: \beta>0$ (correct answer)
$H_0: b=0$ vs. $H_a: b>0$
$H_0: \beta\ne 0$ vs. $H_a: \beta=0$
$H_0: \beta=0$ vs. $H_a: \beta\ne 0$

Explanation: The botanist claims additional sunlight increases plant height, indicating a positive slope in the population regression model. This directional claim requires a one-sided test with the alternative hypothesis showing β > 0. Choice B correctly sets up H₀: β = 0 (no linear relationship) versus Hₐ: β > 0 (positive relationship), using the population parameter β. Choice C incorrectly uses sample slope b, while choice A uses correlation r. Choice E would be appropriate for a two-sided test but doesn't match the directional claim. When a researcher makes a specific directional claim about increase or decrease, the alternative hypothesis must reflect that direction using the appropriate inequality.

Question 19

A school administrator samples $n=60$ students and fits a least-squares regression model predicting final exam score ( $y$ , points) from number of absences ( $x$ ). The administrator's claim is that, in the population, absences are associated with lower exam scores (negative slope). Which hypotheses are appropriate for testing this claim about the population slope?

$H_0: \beta=0$ vs. $H_a: \beta<0$ (correct answer)
$H_0: r=0$ vs. $H_a: r<0$
$H_0: b=0$ vs. $H_a: b<0$
$H_0: \beta=0$ vs. $H_a: \beta>0$
$H_0: \beta<0$ vs. $H_a: \beta\ge 0$

Explanation: The administrator claims absences are associated with lower exam scores, indicating a negative slope relationship. This requires testing H₀: β = 0 (no relationship) against Hₐ: β < 0 (negative relationship). Choice A correctly uses the population slope parameter β with the appropriate one-sided alternative. Choice C incorrectly uses sample slope b instead of population parameter β. Choice B uses correlation r rather than slope. In regression slope testing, we always test claims about the population parameter β, and the alternative hypothesis direction must match the research claim. A claim about "lower" or "reduced" outcomes with increasing x indicates a negative slope.

Question 20

A sports scientist records $n=32$ runners and fits a regression model predicting 5K time ( $y$ , minutes) from weekly training mileage ( $x$ ). The scientist claims that, in the population, more training mileage tends to reduce 5K time (negative slope). Which hypotheses are appropriate for testing this claim about the slope parameter?

$H_0: \beta=0$ vs. $H_a: \beta<0$ (correct answer)
$H_0: r=0$ vs. $H_a: r<0$
$H_0: b=0$ vs. $H_a: b<0$
$H_0: \beta=0$ vs. $H_a: \beta\ne 0$
$H_0: \beta<0$ vs. $H_a: \beta\ge 0$

Explanation: The scientist claims more training mileage reduces 5K time, indicating a negative slope (as mileage increases, time decreases). This requires H₀: β = 0 versus Hₐ: β < 0. Choice A correctly uses the population slope parameter β with the negative alternative hypothesis matching the claim. Choice C incorrectly uses sample slope b, and choice B uses correlation r. Choice D would be for a two-sided test without directional claim. In performance contexts, be careful with direction - "reducing time" or "improving performance" often indicates a negative slope when time is the y-variable. Always test population parameters (β) not sample statistics (b).