Slope of a Regression Model (Test)
Help Questions
AP Statistics › Slope of a Regression Model (Test)
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 that sleep hours predict screen time, but not that screen time predicts sleep hours.
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 of a linear association between screen time and sleep hours in the population.
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.
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.
Because $p=0.049$, there is a 4.9% chance that the slope in this sample is negative.
There is not enough evidence of a linear relationship because the p-value is close to 0.05.
There is convincing evidence that absences cause exam scores to change, so reducing absences will raise every student’s score.
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.
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 of a linear association between fiber intake and LDL cholesterol in the population.
Failing to reject $H_0$ means the slope is exactly 0 for all adults.
There is convincing evidence that higher fiber intake causes LDL cholesterol to decrease.
There is not convincing evidence of a linear association between fiber intake and LDL cholesterol in the population.
Because $p=0.20$, the probability the null hypothesis is true is 0.20.
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.
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.
Failing to reject $H_0$ proves that distance and rent have no relationship of any kind.
Because $p=0.62$, there is a 62% chance the slope is exactly 0 in the sample.
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.
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 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.
There is convincing evidence that weekly sales predict advertising spending, so sales should be the explanatory variable.
There is convincing evidence of a positive linear association between advertising spending and weekly sales in the population of weeks like these.
Because $p=0.008$, there is a 0.8% chance that the slope in the sample is positive.
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.
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 not enough evidence of a linear relationship because the p-value is less than 0.05.
Because $p=0.018$, there is a 1.8% chance that $H_0$ is true.
There is convincing evidence that sleep hours and quiz score are linearly associated in the population.
There is convincing evidence that more sleep causes higher quiz scores for all students.
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.
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 number of bird species predicts habitat area, so the regression should switch the variables.
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 of a linear association between habitat area and number of bird species in the population.
There is convincing evidence that increasing habitat area will cause the number of bird species to increase in every habitat.
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.
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?
Failing to reject $H_0$ means the true slope is positive.
Because $p=0.11$, there is an 11% chance that $H_0$ is true.
There is not convincing evidence of a negative linear relationship between temperature and dissolved oxygen in the population.
There is convincing evidence of a negative linear association between temperature and dissolved oxygen in the population.
There is convincing evidence that higher temperatures cause dissolved oxygen to decrease.
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.
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?
Fail to reject $H_0$; there is not convincing evidence of a linear relationship between sunlight exposure and plant height.
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.
Reject $H_0$; there is convincing evidence of a positive linear relationship because the p-value is close to 0.05.
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.
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 $H_0$; we can be certain that each additional hour of sunlight increases growth by exactly the estimated slope for every plant.
Reject $H_0$; therefore, increasing growth causes plants to receive more sunlight.
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 $H_0$; there is just enough evidence at the 0.05 level to conclude the population slope differs from 0 (a positive linear association).
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.