Slope of a Regression Model (Test) - AP Statistics

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?
Answer: 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. 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.
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?
Answer: 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. 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.
A meteorologist investigates whether atmospheric pressure ( $x$ ) predicts wind speed ( $y$ ) on 28 randomly selected days. The meteorologist fits a linear regression of wind speed on pressure and performs a slope test with $H_0: \beta_1=0$ versus $H_a: \beta_1\neq 0$ . The two-sided p-value is 0.051. Using $\alpha=0.05$ , the meteorologist fails to reject $H_0$ . What conclusion is appropriate?
Answer: There is not convincing evidence of a linear association between pressure and wind speed in the population. This question tests understanding of p-values very close to the significance level. The p-value (0.051) is slightly 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 pressure and wind speed in the population. Choice A incorrectly states we should reject H₀. Choice C incorrectly implies causation. Choice D misinterprets the p-value. Choice E overstates the conclusion—failing to reject H₀ doesn't prove the slope is exactly 0.
A coach wants to know whether practice time ( $x$ , hours per week) predicts free-throw percentage ( $y$ ) for players on a large team league. Using data from 18 randomly selected players, the coach fits a regression of $y$ on $x$ and performs a slope test: $H_0: \beta_1=0$ vs. $H_a: \beta_1>0$ . The one-sided p-value reported is 0.041. At $\alpha=0.05$ , the coach rejects $H_0$ . What conclusion is appropriate?
Answer: There is convincing evidence of a positive linear association between practice time and free-throw percentage in the population. This question involves a one-sided slope test with H₀: β₁ = 0 versus Hₐ: β₁ > 0. The p-value (0.041) is less than α (0.05), so we reject H₀. This provides convincing evidence of a positive linear association between practice time and free-throw percentage in the population. Choice B incorrectly claims causation and overstates the effect. Choice C contradicts the correct decision to reject H₀. Choice D suggests a negative association, which contradicts the positive alternative hypothesis. Choice E misinterprets the p-value as the probability that Hₐ is true.
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?
Answer: Reject $H_0$ ; there is convincing evidence of a positive linear association between square footage and sale price for similar homes. 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.
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?
Answer: Fail to reject $H_0$ ; there is not convincing evidence of a linear relationship between sunlight exposure and plant height. 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.
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?
Answer: There is not convincing evidence of a negative linear relationship between temperature and dissolved oxygen in the population. 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.
A teacher surveyed 28 students and recorded the number of practice problems completed ( $x$ ) and score on a quiz out of 100 ( $y$ ). A least-squares regression of $y$ on $x$ was computed. The slope was tested using $H_0: \beta_1=0$ versus $H_a: \beta_1\neq 0$ , and the p-value was 0.003 with a positive estimated slope. Using $\alpha=0.05$ , the teacher wants to interpret the result. What conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence of a positive linear association between problems completed and quiz score for similar students. This question examines interpretation of a clearly significant result. With p-value = 0.003 << 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 practice problems completed and quiz scores. Choice C wrongly claims causation and universality - the conclusion applies to similar students, not all students. Choice D misinterprets the p-value meaning. Choice E illogically reverses the direction of association. When we reject H₀ with a small p-value and positive slope, we have strong evidence that completing more problems is associated with higher quiz scores in the population of similar students.
A marketing analyst examined whether ad spending relates to sales using $n=12$ months of data: monthly online ad spending ( $x$ , in thousands of dollars) and monthly sales revenue ( $y$ , in thousands of dollars). A least-squares regression of sales on ad spending produced slope $b_1=1.7$ . The analyst tested $H_0: \beta_1=0$ versus $H_a: \beta_1\ne 0$ and obtained a p-value of $0.049$ . At the 0.05 level, what conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the true slope differs from 0, so ad spending and sales are linearly associated for this business over the months studied. This question involves a two-sided slope test where the p-value (0.049) is less than α = 0.05. Since 0.049 < 0.05, we reject H₀: β₁ = 0. This provides convincing evidence that the true slope differs from 0, meaning ad spending and sales are linearly associated. Choice B correctly states this conclusion. Choice A incorrectly suggests the sample size affects the p-value threshold - we use α = 0.05 regardless. Choice C makes an inappropriate causal claim. Choice D misinterprets the p-value as the probability the alternative is false. Choice E reverses the direction of causation. When p-value < α in a two-sided test, we reject H₀ and conclude there is evidence of a linear relationship between the variables.
A city planner collected data from $n=32$ intersections on average daily traffic volume (in thousands of cars, $x$ ) and average wait time at the light (in seconds, $y$ ). A least-squares regression of wait time on traffic volume produced slope $b_1=2.4$ seconds per 1,000 cars. To test whether there is a linear relationship, the planner performed a slope test with hypotheses $H_0: \beta_1=0$ versus $H_a: \beta_1\ne 0$ at $\alpha=0.05$ and obtained a p-value of $0.018$ . Based on this completed test, what conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the true slope is not 0, so traffic volume is linearly associated with average wait time at intersections in this city. This question tests understanding of slope test conclusions when the p-value (0.018) is less than the significance level (0.05). Since 0.018 < 0.05, we reject the null hypothesis that the true slope β₁ = 0. This provides convincing evidence that there is a linear relationship between traffic volume and wait time. Choice A incorrectly interprets the p-value as the probability that H₀ is true, when it actually represents the probability of getting data this extreme if H₀ were true. Choice B correctly states we reject H₀ and concludes there is a linear association. Choices D and E make causal claims that cannot be supported by observational data alone. When we reject H₀ in a slope test, we conclude there is evidence of a linear relationship between the variables, not that one causes the other.
A teacher recorded $n=40$ students' number of hours studied for a final exam ( $x$ ) and their exam scores ( $y$ ). A least-squares regression of score on hours studied produced slope $b_1=5.8$ points per hour. The teacher tested $H_0: \beta_1=0$ versus $H_a: \beta_1>0$ and obtained a p-value of $0.0006$ . Using this completed slope test, what conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the true slope is positive, so exam score is positively linearly associated with hours studied for this group of students. This problem tests interpretation of a one-sided slope test where the p-value (0.0006) is much less than α = 0.05. Since 0.0006 < 0.05, we reject H₀: β₁ = 0 in favor of Hₐ: β₁ > 0. This provides convincing evidence that the true slope is positive, meaning exam scores tend to increase linearly with hours studied. Choice B correctly states this conclusion about a positive linear association. Choice C makes an inappropriate causal claim - regression analysis alone cannot prove causation. Choice D misinterprets the p-value, and Choice E reverses the direction of causation. Choice A incorrectly fails to reject H₀ despite the small p-value. When we reject H₀ in a slope test, we conclude there is evidence of the specified linear relationship, not causation.
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?
Answer: Fail to reject $H_0$ ; there is not convincing evidence of a linear relationship between days since treatment and algae concentration. 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.
A real estate analyst collected data from $n=22$ homes on size ( $x$ , in square feet) and sale price ( $y$ , in thousands of dollars). The least-squares regression of price on size produced slope $b_1=0.19$ . The analyst tested $H_0: \beta_1=0$ versus $H_a: \beta_1>0$ and obtained a p-value of $0.047$ . Using this completed slope test at $\alpha=0.05$ , what conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the true slope is positive, so larger homes tend to have higher sale prices in a linear way in the market studied. This problem tests interpretation of a one-sided slope test where Hₐ: β₁ > 0. The p-value of 0.047 is less than α = 0.05 (though just barely), so we reject H₀: β₁ = 0. This provides convincing evidence that the true slope is positive, meaning larger homes tend to have higher sale prices in a linear way. Choice B correctly states this conclusion. Choice A incorrectly fails to reject H₀ - being close to 0.05 doesn't matter; 0.047 < 0.05 means we reject. Choice C makes an inappropriate causal claim. Choice D reverses the direction of causation. Choice E misinterprets the p-value as the probability the alternative is true. When p-value < α, we reject H₀ regardless of how close the values are.
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?
Answer: There is not convincing evidence of a linear association between fiber intake and LDL cholesterol in the population. 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.
A city planner sampled 35 intersections and recorded average daily traffic volume in thousands of cars ( $x$ ) and the number of accidents in a year ( $y$ ). A linear regression of $y$ on $x$ was fit. A test of the slope used $H_0: \beta_1=0$ versus $H_a: \beta_1\neq 0$ , and the p-value was 0.020 with a positive estimated slope. At the 0.05 level, the planner wants to make a conclusion about linear association. What conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence of a positive linear association between traffic volume and number of accidents at similar intersections. This question tests interpretation of a significant slope test result. With p-value = 0.020 < α = 0.05, we reject H₀: β₁ = 0. The positive estimated slope means we have convincing evidence of a positive linear association between traffic volume and accidents. Choice C incorrectly claims causation and certainty about intervention effects. Choice D misinterprets the p-value - it's the probability of observing data this extreme if H₀ were true, not the probability the slope is positive. Choice E reverses causation illogically. When we reject H₀ with p < α and observe a positive slope estimate, we conclude there is convincing evidence of a positive linear association in the population of similar intersections.
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?
Answer: 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). 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.
A marketing team sampled 28 stores and recorded amount spent on local advertising ( $x$ ) and weekly revenue ( $y$ ). A least-squares regression of $y$ on $x$ was fit. The slope test used $H_0: \beta_1=0$ vs. $H_a: \beta_1\ne 0$ and produced a p-value of 0.003 with a positive estimated slope, so the team rejected $H_0$ at $\alpha=0.05$ . What conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence of a positive linear association between advertising spending and revenue in the population of similar stores. This question evaluates conclusions from a significant slope test in regression analysis. The p-value of 0.003 < α=0.05 leads to rejecting H0: β1=0, providing convincing evidence of a positive linear association between advertising spending and revenue. The positive slope suggests higher spending links to higher revenue. Choice C is a distractor, wrongly claiming causation from observational data. Mini-lesson: Rejecting H0 indicates evidence against a zero slope, supporting population association, but causation requires experimental design. Small p-values denote strong evidence, but always specify the direction based on the estimate.
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?
Answer: Reject $H_0$ ; there is convincing evidence of a nonzero linear relationship between practice problems completed and quiz score in the population. 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.
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?
Answer: There is convincing evidence of a linear association between absences and final exam score in the population. 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.
A fitness app company sampled 30 users and recorded average daily steps ( $x$ ) and resting heart rate ( $y$ ). A least-squares regression of $y$ on $x$ was computed. The slope test used $H_0: \beta_1=0$ vs. $H_a: \beta_1\ne 0$ and gave a p-value of 0.078 with an estimated negative slope. At the $\alpha=0.05$ level, the company failed to reject $H_0$ . What conclusion is appropriate?
Answer: Fail to reject $H_0$ ; there is not convincing evidence at the 0.05 level that the population slope differs from 0. This question assesses interpreting a slope hypothesis test in linear regression, particularly when the p-value exceeds the significance level. The p-value of 0.078 is greater than α=0.05, so we fail to reject H0: β1=0, meaning there is not convincing evidence of a linear association between daily steps and resting heart rate. The negative slope implies that more steps might relate to lower heart rates, but the evidence is not strong enough at this alpha level. A frequent distractor is choice A, which wrongly suggests rejecting H0 despite the p-value being above 0.05. Mini-lesson on conclusions: Rejecting H0 supports a nonzero slope and linear association, but failing to reject means insufficient evidence—it does not prove the slope is exactly zero. Consider sample size and power, as a larger sample might detect smaller effects.

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