AP Statistics Quiz: Slope Of A Regression Model Test

20 questions · exam conditions

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Slope Of A Regression Model TestQuestion 1 of 20

A researcher examines whether daily screen time ( $x$ , hours) predicts number of hours slept ( $y$ ) for 40 randomly selected adults. A regression of sleep on screen time is fit, and a slope test is carried out with $H_0: \beta_1=0$ versus $H_a: \beta_1\neq 0$ . The output reports a two-sided p-value of 0.003. At $\alpha=0.01$ , the researcher rejects $H_0$ . What conclusion is appropriate?

There is convincing evidence that screen time causes changes in sleep hours.

There is convincing evidence of a linear association between screen time and sleep hours in the population.

There is not convincing evidence of a linear association because 0.003 is less than 0.01.

Because

p=0.003

, the probability that the true slope equals 0 is 0.003.

There is convincing evidence that sleep hours predict screen time, but not that screen time predicts sleep hours.

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AP Statistics Quiz

AP Statistics Quiz: Slope Of A Regression Model Test

Practice Slope Of A Regression Model Test 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 Test, 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

There is convincing evidence that screen time causes changes in sleep hours.
There is convincing evidence of a linear association between screen time and sleep hours in the population. (correct answer)
There is not convincing evidence of a linear association because 0.003 is less than 0.01.
Because $p=0.003$ , the probability that the true slope equals 0 is 0.003.
There is convincing evidence that sleep hours predict screen time, but not that screen time predicts sleep hours.

Explanation: This question tests understanding of slope test conclusions with a stringent significance level. The p-value (0.003) is less than α (0.01), so we reject H₀: β₁ = 0. This provides convincing evidence of a linear association between screen time and sleep hours in the population. Choice A incorrectly implies causation from an observational study. Choice C misunderstands the decision rule—we reject H₀ when p < α. Choice D misinterprets the p-value as the probability that the true slope equals 0. Choice E incorrectly suggests the direction of prediction matters for the association conclusion.

Question 2

A teacher investigates whether number of absences ( $x$ ) predicts final exam score ( $y$ ) for 22 students and fits a least-squares regression of score on absences. A slope test is performed with $H_0: \beta_1=0$ versus $H_a: \beta_1\neq 0$ , and the two-sided p-value is reported as 0.049. Using $\alpha=0.05$ , the teacher rejects $H_0$ . What conclusion is appropriate?

There is convincing evidence of a linear association between absences and final exam score in the population. (correct answer)
There is convincing evidence that absences cause exam scores to change, so reducing absences will raise every student's score.
There is not enough evidence of a linear relationship because the p-value is close to 0.05.
Because $p=0.049$ , there is a 4.9% chance that the slope in this sample is negative.
There is convincing evidence that exam score predicts absences, so absences should be treated as the response variable.

Explanation: This question tests understanding of borderline p-values in slope tests. The p-value (0.049) is just barely less than α (0.05), so we reject H₀: β₁ = 0. This provides convincing evidence of a linear association between absences and final exam score in the population. Choice B incorrectly implies causation and overstates the effect. Choice C misunderstands the decision rule—we reject H₀ when p < α, even if barely. Choice D misinterprets what the p-value represents. Choice E incorrectly suggests switching the predictor and response variables based on the test result.

Question 3

A nutritionist studies whether daily fiber intake ( $x$ , grams) predicts LDL cholesterol ( $y$ , mg/dL) using a random sample of 35 adults. The regression of LDL on fiber is fit, and a slope test is conducted: $H_0: \beta_1=0$ vs. $H_a: \beta_1\neq 0$ . The two-sided p-value is 0.20, so at $\alpha=0.05$ the nutritionist fails to reject $H_0$ . What conclusion is appropriate?

There is convincing evidence that higher fiber intake causes LDL cholesterol to decrease.
There is convincing evidence of a linear association between fiber intake and LDL cholesterol in the population.
There is not convincing evidence of a linear association between fiber intake and LDL cholesterol in the population. (correct answer)
Because $p=0.20$ , the probability the null hypothesis is true is 0.20.
Failing to reject $H_0$ means the slope is exactly 0 for all adults.

Explanation: This question asks about interpreting a slope test when we fail to reject H₀. The p-value (0.20) is much greater than α (0.05), so we fail to reject H₀: β₁ = 0. This means there is not convincing evidence of a linear association between fiber intake and LDL cholesterol in the population. Choice A incorrectly implies causation. Choice B would be correct if we had rejected H₀. Choice D misinterprets the p-value as the probability that H₀ is true. Choice E overstates the conclusion—failing to reject H₀ doesn't prove the slope is exactly 0.

Question 4

A city planner studies whether distance from downtown ( $x$ , in miles) predicts monthly rent ( $y$ , in dollars) for a random sample of 25 apartments. A slope test is conducted for the regression of rent on distance with $H_0: \beta_1=0$ and $H_a: \beta_1\neq 0$ . The output gives a two-sided p-value of 0.62 for the slope. At the 0.05 level, the planner fails to reject $H_0$ . What conclusion is appropriate?

There is convincing evidence that distance from downtown is linearly related to monthly rent in the population.
There is not convincing evidence of a linear relationship between distance from downtown and monthly rent in the population. (correct answer)
Because $p=0.62$ , there is a 62% chance the slope is exactly 0 in the sample.
Failing to reject $H_0$ proves that distance and rent have no relationship of any kind.
There is convincing evidence that higher rent causes apartments to be farther from downtown.

Explanation: This question asks about interpreting a slope test when we fail to reject the null hypothesis. The p-value (0.62) is much larger than the significance level (0.05), so we fail to reject H₀: β₁ = 0. This means there is not convincing evidence of a linear relationship between distance from downtown and monthly rent in the population. Choice A would be correct if we had rejected H₀. Choice C misinterprets the p-value. Choice D overstates the conclusion—failing to reject H₀ doesn't prove no relationship exists. Choice E incorrectly implies causation.

Question 5

A business analyst tests whether advertising spending ( $x$ , thousands of dollars) predicts weekly sales ( $y$ , thousands of dollars) for 16 weeks. A regression of sales on ad spending is fit, and a slope test is performed with $H_0: \beta_1=0$ versus $H_a: \beta_1>0$ . The reported one-sided p-value is 0.008, and at $\alpha=0.05$ the analyst rejects $H_0$ . What conclusion is appropriate?

There is convincing evidence of a positive linear association between advertising spending and weekly sales in the population of weeks like these. (correct answer)
There is convincing evidence that increasing advertising spending will cause weekly sales to increase.
There is not convincing evidence of a positive linear association because 0.008 is less than 0.05.
Because $p=0.008$ , there is a 0.8% chance that the slope in the sample is positive.
There is convincing evidence that weekly sales predict advertising spending, so sales should be the explanatory variable.

Explanation: This question involves a one-sided test with Hₐ: β₁ > 0. The p-value (0.008) is less than α (0.05), so we reject H₀. This provides convincing evidence of a positive linear association between advertising spending and weekly sales in the population of weeks like these. Choice B incorrectly implies causation from observational data. Choice C misunderstands the decision rule. Choice D misinterprets what the p-value represents. Choice E incorrectly suggests switching variables based on the test result.

Question 6

A student investigates whether hours of sleep ( $x$ ) predicts quiz score ( $y$ ) for 30 classmates and fits the least-squares regression line. A slope test is performed with hypotheses $H_0: \beta_1=0$ versus $H_a: \beta_1\neq 0$ . The computer output reports a two-sided p-value of 0.018 for the slope. Using a 0.05 significance level, the student decides to reject $H_0$ . Based on this completed slope test, what conclusion is appropriate?

There is convincing evidence that sleep hours and quiz score are linearly associated in the population. (correct answer)
There is convincing evidence that more sleep causes higher quiz scores for all students.
Because $p=0.018$ , there is a 1.8% chance that $H_0$ is true.
There is not enough evidence of a linear relationship because the p-value is less than 0.05.
There is convincing evidence that quiz score predicts hours of sleep in the population.

Explanation: This question tests understanding of slope test conclusions in linear regression. Since the p-value (0.018) is less than the significance level (0.05), we reject the null hypothesis that the slope equals zero. This provides convincing evidence of a linear association between sleep hours and quiz score in the population. Choice B incorrectly implies causation, which cannot be established from an observational study. Choice C misinterprets the p-value as the probability that H₀ is true. Choice D contradicts the correct decision to reject H₀. Choice E reverses the predictor and response variables.

Question 7

A wildlife biologist studies whether habitat area ( $x$ , acres) predicts number of bird species observed ( $y$ ) in a region. Using data from 14 randomly selected habitats, the biologist fits a least-squares regression of species count on area and conducts a slope test: $H_0: \beta_1=0$ versus $H_a: \beta_1\neq 0$ . The two-sided p-value for the slope is 0.0006, so at $\alpha=0.05$ the biologist rejects $H_0$ . What conclusion is appropriate?

There is convincing evidence that increasing habitat area will cause the number of bird species to increase in every habitat.
There is convincing evidence of a linear association between habitat area and number of bird species in the population. (correct answer)
There is not convincing evidence of a linear association because the p-value is very small.
Because $p=0.0006$ , there is a 0.06% chance the alternative hypothesis is false.
There is convincing evidence that number of bird species predicts habitat area, so the regression should switch the variables.

Explanation: This question involves a very small p-value in a slope test. The p-value (0.0006) is much less than α (0.05), so we reject H₀: β₁ = 0. This provides convincing evidence of a linear association between habitat area and number of bird species in the population. Choice A incorrectly implies causation and overstates the effect. Choice C misunderstands that small p-values lead to rejecting H₀. Choice D misinterprets what the p-value represents. Choice E incorrectly suggests switching the predictor and response based on the test result.

Question 8

An environmental scientist studies whether water temperature ( $x$ , in °C) predicts dissolved oxygen ( $y$ , in mg/L) in a river. Using 12 measurements taken on randomly selected days, the scientist fits a linear regression of $y$ on $x$ and tests $H_0: \beta_1=0$ versus $H_a: \beta_1<0$ . The output gives a one-sided p-value of 0.11. At $\alpha=0.05$ , the scientist fails to reject $H_0$ . What conclusion is appropriate?

There is convincing evidence that higher temperatures cause dissolved oxygen to decrease.
There is convincing evidence of a negative linear association between temperature and dissolved oxygen in the population.
There is not convincing evidence of a negative linear relationship between temperature and dissolved oxygen in the population. (correct answer)
Because $p=0.11$ , there is an 11% chance that $H_0$ is true.
Failing to reject $H_0$ means the true slope is positive.

Explanation: This question involves a one-sided test with Hₐ: β₁ < 0 (testing for a negative slope). The p-value (0.11) is greater than α (0.05), so we fail to reject H₀. This means there is not convincing evidence of a negative linear relationship between temperature and dissolved oxygen in the population. Choice A incorrectly implies causation. Choice B would be correct if we had rejected H₀. Choice D misinterprets the p-value. Choice E incorrectly suggests that failing to reject H₀ means the slope must be positive.

Question 9

A botanist measured sunlight exposure in hours per day ( $x$ ) and plant height in centimeters ( $y$ ) for 16 plants of the same species grown in a greenhouse. A least-squares regression of $y$ on $x$ was fit and a slope test was conducted: $H_0: \beta_1=0$ versus $H_a: \beta_1\neq 0$ . The estimated slope was positive with p-value 0.067. Using $\alpha=0.05$ , the botanist wants to interpret the slope test.

What conclusion is appropriate?

Reject $H_0$ ; there is convincing evidence of a positive linear relationship because the p-value is close to 0.05.
Fail to reject $H_0$ ; there is not convincing evidence of a linear relationship between sunlight exposure and plant height. (correct answer)
Fail to reject $H_0$ ; this proves the true slope is exactly 0.
Reject $H_0$ ; additional sunlight causes plants to grow taller because the estimated slope is positive.
Reject $H_0$ ; there is convincing evidence that plant height causes sunlight exposure to increase.

Explanation: This question assesses understanding of p-values slightly above the significance level. With p-value = 0.067 > α = 0.05, we fail to reject H₀: β₁ = 0. Even though the p-value is relatively close to 0.05 and the estimated slope is positive, we cannot conclude there is convincing evidence of a linear relationship. Choice A incorrectly rejects H₀ when p > α. Choice C overstates the conclusion - failing to reject never proves H₀ is true. Choice D wrongly infers causation. When p-value > α in a slope test, regardless of how close it is to α, we conclude there is not convincing evidence of a linear relationship between the variables.

Question 10

A biologist studied 12 plants and measured hours of sunlight per day ( $x$ ) and weekly growth ( $y$ in cm). A least-squares regression of $y$ on $x$ was fit and the slope test used $H_0: \beta_1=0$ vs. $H_a: \beta_1\ne 0$ . The p-value was 0.049 with a positive estimated slope. Using $\alpha=0.05$ , the biologist rejected $H_0$ . What conclusion is appropriate?

Reject $H_0$ ; there is just enough evidence at the 0.05 level to conclude the population slope differs from 0 (a positive linear association). (correct answer)
Fail to reject $H_0$ ; because the sample size is only 12, the test cannot be used.
Reject $H_0$ ; the p-value 0.049 means there is a 4.9% chance that the true slope is 0.
Reject $H_0$ ; therefore, increasing growth causes plants to receive more sunlight.
Reject $H_0$ ; we can be certain that each additional hour of sunlight increases growth by exactly the estimated slope for every plant.

Explanation: This question assesses interpreting a borderline significant p-value in a slope test for linear regression. With p=0.049 just below α=0.05, we reject H0: β1=0, concluding there is evidence (albeit marginal) of a positive linear association between sunlight and plant growth. The positive slope means more sunlight associates with greater growth. Choice C misleads by misinterpreting the p-value as the probability of the slope being zero, but it's the probability of data under H0. Mini-lesson: The decision hinges on comparing p to alpha; rejection supports a nonzero slope and association, with the estimate's sign indicating direction, but 'just enough evidence' reminds us significance is threshold-based. Small samples like n=12 can still yield valid tests if assumptions hold.

Question 11

A real estate analyst sampled 22 homes and recorded square footage ( $x$ ) and sale price in thousands of dollars ( $y$ ). A linear regression of $y$ on $x$ was fit, and the slope was tested with $H_0: \beta_1=0$ versus $H_a: \beta_1\neq 0$ . The estimated slope was positive and the p-value was $<0.001$ . Using $\alpha=0.05$ , the analyst wants to report an appropriate conclusion.

What conclusion is appropriate?

Fail to reject $H_0$ ; a small p-value indicates the slope could easily be 0.
Reject $H_0$ ; there is convincing evidence of a positive linear association between square footage and sale price for similar homes. (correct answer)
Reject $H_0$ ; increasing square footage causes sale price to increase for every home because the p-value is extremely small.
Reject $H_0$ ; there is convincing evidence that sale price causes square footage to be larger.
Fail to reject $H_0$ ; there is not convincing evidence of a linear relationship because the sample is not huge.

Explanation: This question tests interpretation of a highly significant slope test. With p-value < 0.001, which is much less than α = 0.05, we strongly reject H₀: β₁ = 0. Combined with the positive estimated slope, this provides convincing evidence of a positive linear association between square footage and sale price. Choice C incorrectly claims causation and universality - we can only conclude association for similar homes. Choice D reverses the direction of potential causation. Choice E wrongly suggests the sample size invalidates the result. When we reject H₀ with a very small p-value and positive slope estimate, we have strong evidence of a positive linear association in the population of similar units.

Question 12

A meteorologist recorded $n=15$ days of data: morning humidity ( $x$ , percent) and afternoon high temperature ( $y$ , degrees Fahrenheit). A least-squares regression of temperature on humidity produced slope $b_1=-0.22$ . The meteorologist tested $H_0: \beta_1=0$ versus $H_a: \beta_1\ne 0$ and obtained a p-value of $0.62$ . At $\alpha=0.05$ , what conclusion is appropriate?

Reject $H_0$ ; there is convincing evidence of a nonzero linear relationship between humidity and temperature.
Fail to reject $H_0$ ; the data do not provide convincing evidence that the true slope differs from 0 for the relationship between humidity and temperature. (correct answer)
Fail to reject $H_0$ ; therefore, humidity and temperature are independent.
Because the p-value is 0.62, there is a 62% chance that $H_0$ is false.
Fail to reject $H_0$ ; therefore, higher temperatures cause lower humidity.

Explanation: This question involves a two-sided slope test where the p-value (0.62) is much larger than α = 0.05. Since 0.62 > 0.05, we fail to reject H₀: β₁ = 0. This means the data do not provide convincing evidence that the true slope differs from 0 for the relationship between humidity and temperature. Choice B correctly states this conclusion. Choice A incorrectly rejects H₀ when the p-value is too large. Choice C incorrectly claims the variables are independent - failing to reject H₀ doesn't prove independence. Choice D misinterprets the p-value as the probability H₀ is false. Choice E makes an inappropriate causal claim. When the p-value > α in a slope test, we conclude there is insufficient evidence of a linear relationship.

Question 13

An environmental scientist recorded the number of days since a lake was treated ( $x$ ) and the algae concentration ( $y$ ) for 18 lakes. A least-squares regression of $y$ on $x$ was computed, and a test of the slope was conducted: $H_0: \beta_1=0$ versus $H_a: \beta_1\neq 0$ . The output showed a positive estimated slope with p-value 0.41. At the 0.05 level, the scientist wants to decide whether there is evidence of a linear relationship between days since treatment and algae concentration.

What conclusion is appropriate?

Reject $H_0$ ; there is convincing evidence of a positive linear relationship between days since treatment and algae concentration.
Fail to reject $H_0$ ; there is not convincing evidence of a linear relationship between days since treatment and algae concentration. (correct answer)
Fail to reject $H_0$ ; the p-value 0.41 proves there is no relationship of any kind between the variables.
Reject $H_0$ ; days since treatment causes algae concentration to increase because the slope is positive.
Reject $H_0$ ; there is convincing evidence that algae concentration causes more days to pass since treatment.

Explanation: This question assesses interpretation of a non-significant slope test result. The p-value of 0.41 is much larger than the significance level of 0.05, so we fail to reject the null hypothesis H₀: β₁ = 0. This means we do not have convincing evidence that the true slope differs from zero, and therefore cannot conclude there is a linear relationship between days since treatment and algae concentration. Choice C incorrectly claims this proves no relationship exists - failing to reject H₀ never proves H₀ is true. Choice D wrongly infers causation from a positive slope estimate. When p-value > α in a slope test, we conclude there is not convincing evidence of a linear relationship between the variables, which could mean either no relationship exists or our sample was too small to detect it.

Question 14

A teacher recorded data for 16 students on number of practice problems completed ( $x$ ) and score on a quiz ( $y$ ). A least-squares regression of $y$ on $x$ was fit, and a slope test was conducted: $H_0: \beta_1=0$ vs. $H_a: \beta_1\ne 0$ . The p-value was 0.032 and the estimated slope was positive. At $\alpha=0.05$ , the teacher rejected $H_0$ . What conclusion is appropriate?

Reject $H_0$ ; there is convincing evidence of a nonzero linear relationship between practice problems completed and quiz score in the population. (correct answer)
Fail to reject $H_0$ ; the p-value is less than 0.05, so there is not convincing evidence.
Reject $H_0$ ; completing more practice problems causes higher quiz scores for all students.
Reject $H_0$ ; the p-value 0.032 means there is a 3.2% chance the alternative hypothesis is false.
Reject $H_0$ ; therefore, higher quiz scores cause students to complete more practice problems.

Explanation: This question tests understanding of slope hypothesis testing in regression, focusing on appropriate conclusions from rejecting H0. With p=0.032 < α=0.05, we reject H0: β1=0, providing evidence of a linear relationship between practice problems and quiz scores. The positive slope suggests more problems completed associate with higher scores. Choice C is a distractor because it implies causation and universality, but the test only supports association, not cause, and not for all students. Mini-lesson: The slope test evaluates if β1 ≠ 0, supporting population-level linear association if rejected, but does not imply causation or individual predictions. Use the p-value to gauge evidence strength relative to alpha.

Question 15

A real estate analyst examined 22 recently sold homes and recorded living area ( $x$ , square feet) and sale price ( $y$ , thousands of dollars). A least-squares regression of $y$ on $x$ was fit. The slope test $H_0: \beta_1=0$ vs. $H_a: \beta_1\ne 0$ produced a p-value of 0.0006 and a positive estimated slope. Using $\alpha=0.01$ , the analyst rejected $H_0$ . What conclusion is appropriate?

Reject $H_0$ ; there is very strong evidence that the population slope is positive, so larger living area is linearly associated with higher sale price. (correct answer)
Fail to reject $H_0$ ; the p-value is less than 0.01 so the result is not statistically significant.
Reject $H_0$ ; the p-value 0.0006 means there is a 0.06% chance that the slope estimate is positive just by chance.
Reject $H_0$ ; increasing sale price causes homes to have more square feet.
Reject $H_0$ ; we can be certain the relationship will hold for every individual home.

Explanation: This question examines hypothesis testing for the slope of a regression line, emphasizing strong evidence from a very small p-value. The p-value of 0.0006 is much less than α=0.01, leading us to reject H0: β1=0 and conclude convincing evidence of a positive linear association between living area and sale price. The positive slope means larger homes tend to sell for more. Choice D is a distractor as it incorrectly claims causation, but observational data like this cannot establish that increasing price causes more square feet. Mini-lesson: A significant slope test indicates the variables are linearly related in the population, with the sign showing the direction, but avoid causal language unless from an experiment. Very small p-values suggest strong evidence against H0.

Question 16

An environmental scientist recorded data from 25 lakes on water temperature ( $x$ , in $^\circ$ C) and dissolved oxygen ( $y$ , mg/L). A least-squares regression of $y$ on $x$ was fit. A test of the slope used $H_0: \beta_1=0$ vs. $H_a: \beta_1\ne 0$ and produced a p-value of 0.41, with an estimated negative slope. Using $\alpha=0.05$ , the scientist failed to reject $H_0$ . What conclusion is appropriate?

Fail to reject $H_0$ ; there is not convincing evidence of a nonzero linear relationship between temperature and dissolved oxygen in the population. (correct answer)
Reject $H_0$ ; since the slope estimate is negative, there must be a negative linear association in the population.
Fail to reject $H_0$ ; therefore, temperature and dissolved oxygen are not related in any way.
Reject $H_0$ ; the p-value 0.41 means there is a 41% chance the alternative hypothesis is true.
Fail to reject $H_0$ ; therefore, higher dissolved oxygen causes lower temperature.

Explanation: This question evaluates knowledge of hypothesis testing for the regression slope, focusing on when to fail to reject the null hypothesis. With a p-value of 0.41 greater than α=0.05, we fail to reject H0: β1=0, indicating insufficient evidence of a linear relationship between temperature and dissolved oxygen in the population. The negative slope estimate suggests a potential inverse association, but the high p-value means we cannot conclude it's statistically significant. Choice C is a distractor because failing to reject H0 does not prove no relationship exists at all—it only means no convincing evidence of a linear one. Mini-lesson: The slope test checks for evidence against β1=0; a non-significant result means we lack evidence to claim a linear association, but other types of relationships might still be possible. Interpret p-values carefully, as they represent the probability of the observed data under H0, not the probability of hypotheses.

Question 17

A school counselor examined whether attendance relates to GPA using data from $n=60$ students: number of absences in a semester ( $x$ ) and GPA ( $y$ ). A least-squares regression of GPA on absences produced slope $b_1=-0.07$ GPA points per absence. The counselor tested $H_0: \beta_1=0$ versus $H_a: \beta_1\ne 0$ and obtained a p-value of $0.29$ . At the 0.05 level, what conclusion is appropriate?

Reject $H_0$ ; there is convincing evidence that absences and GPA are linearly associated because the slope is negative.
Fail to reject $H_0$ ; there is not convincing evidence of a linear relationship between absences and GPA in the population of students like those studied. (correct answer)
Fail to reject $H_0$ ; therefore, absences do not affect GPA and the true slope is 0.
Because the p-value is 0.29, there is a 29% chance that $H_0$ is true.
Fail to reject $H_0$ ; this shows that higher GPA causes fewer absences.

Explanation: This question involves a two-sided slope test where the p-value (0.29) is much larger than α = 0.05. Since 0.29 > 0.05, we fail to reject H₀: β₁ = 0. This means the data do not provide convincing evidence of a linear relationship between absences and GPA in the population of students studied. Choice B correctly states this conclusion. Choice A incorrectly rejects H₀ when the p-value is too large. Choice C incorrectly claims this proves the true slope is 0 - failing to reject H₀ never proves H₀ is true. Choice D misinterprets the p-value as the probability that H₀ is true. Choice E makes an inappropriate causal claim. When the p-value exceeds α, we fail to reject H₀ and conclude there is insufficient evidence for a linear relationship.

Question 18

A public health researcher studied $n=50$ adults and recorded weekly minutes of exercise ( $x$ ) and resting heart rate ( $y$ , beats per minute). A least-squares regression of heart rate on exercise produced slope $b_1=-0.05$ . The researcher tested $H_0: \beta_1=0$ versus $H_a: \beta_1<0$ and obtained a p-value of $0.0019$ . Based on this completed slope test, what conclusion is appropriate?

Because the p-value is 0.0019, there is a 0.19% chance that the null hypothesis is true.
Reject $H_0$ ; there is convincing evidence that the true slope is negative, so greater weekly exercise is associated with lower resting heart rate in a linear way in the population studied. (correct answer)
Fail to reject $H_0$ ; there is not enough evidence of a negative linear relationship.
Reject $H_0$ ; this proves that increasing exercise causes resting heart rate to decrease for all adults.
Reject $H_0$ ; therefore, lower resting heart rate causes people to exercise more minutes per week.

Explanation: This problem tests understanding of a one-sided slope test where Hₐ: β₁ < 0. The p-value of 0.0019 is much less than α = 0.05, so we reject H₀: β₁ = 0. This provides convincing evidence that the true slope is negative, meaning greater weekly exercise is associated with lower resting heart rate in a linear way. Choice B correctly states this conclusion about the negative linear association. Choice A misinterprets the p-value as the probability H₀ is true. Choice C incorrectly fails to reject H₀ despite the very small p-value. Choice D makes an inappropriate claim about causation and extends beyond the studied population. Choice E reverses the direction of causation. When we reject H₀ in favor of Hₐ: β₁ < 0, we conclude there is evidence of a negative linear relationship.

Question 19

A school counselor collected data from 18 students on weekly hours of sleep ( $x$ ) and number of days absent in a semester ( $y$ ). A least-squares regression of $y$ on $x$ was fit, and a slope test was performed: $H_0: \beta_1=0$ vs. $H_a: \beta_1\ne 0$ . The computer output reported a negative estimated slope and a p-value of 0.012. Using $\alpha=0.05$ , the counselor rejected $H_0$ and concluded there is convincing evidence of a linear association between sleep and absences in the population of similar students. What conclusion is appropriate?

Because the p-value is 0.012, there is a 1.2% chance that $H_0$ is true.
Reject $H_0$ ; there is convincing evidence that the slope is not 0, so sleep hours are linearly associated with absences (with a negative direction) in the population. (correct answer)
Fail to reject $H_0$ ; the p-value is greater than 0.05, so there is not convincing evidence of a linear relationship.
Reject $H_0$ ; increasing absences causes students to sleep fewer hours each week.
Reject $H_0$ ; we can be certain that more sleep will reduce absences for every student.

Explanation: This question tests your understanding of hypothesis testing for the slope in a linear regression model, specifically interpreting the results of a test for whether the population slope β1 is zero. The p-value of 0.012 is less than the significance level α=0.05, so we reject the null hypothesis H0: β1=0, providing convincing evidence of a linear association between sleep hours and absences in the population. The negative estimated slope indicates that as sleep hours increase, absences tend to decrease. A common distractor is choice D, which incorrectly infers causation and reverses the direction, but regression alone does not establish cause-and-effect relationships. In a mini-lesson on slope test conclusions: this test assesses evidence for a nonzero slope in the population, meaning a linear association exists, but it does not prove causation or guarantee the relationship holds for every individual. Always consider the direction of the slope when describing the association.

Question 20

A city planner sampled 40 intersections and recorded average daily traffic volume ( $x$ ) and number of accidents in a year ( $y$ ). A least-squares regression of $y$ on $x$ was computed. The slope test $H_0: \beta_1=0$ vs. $H_a: \beta_1\ne 0$ returned a p-value of 0.19 with a positive estimated slope, so the planner failed to reject $H_0$ at $\alpha=0.05$ . What conclusion is appropriate?

Fail to reject $H_0$ ; there is not convincing evidence of a linear association between traffic volume and accidents in the population. (correct answer)
Reject $H_0$ ; the positive slope estimate proves higher traffic volume increases accidents.
Fail to reject $H_0$ ; therefore, there is no relationship of any kind between traffic volume and accidents.
Reject $H_0$ ; a p-value of 0.19 means there is a 19% chance that $H_0$ is false.
Fail to reject $H_0$ ; therefore, more accidents cause higher traffic volume.

Explanation: This question probes knowledge of failing to reject the null in a regression slope test. The p-value of 0.19 > α=0.05 means we fail to reject H0: β1=0, so there is not convincing evidence of a linear association between traffic volume and accidents. The positive slope hints at more traffic potentially linking to more accidents, but it's not statistically significant. Choice B distracts by suggesting rejection and causation, ignoring the p-value decision. Mini-lesson: Failing to reject H0 indicates lack of evidence for a nonzero slope, but does not disprove any association—consider nonlinear relationships or larger samples for more power. Avoid overinterpreting non-significance as proof of no effect.