Justifying Claims: Slope of Regression Models
Help Questions
AP Statistics › Justifying Claims: Slope of Regression Models
A sports scientist models a runner’s 5K time (minutes) from the number of weeks in a training program. The scientist claims that training reduces 5K time by between 0.10 and 0.30 minutes per week. A 95% confidence interval for the slope is $(-0.28, -0.12)$ minutes per week. Is the claim supported by the confidence interval?
No, because the interval is negative, so it contradicts any claim about reduction.
No, because the interval does not include 0, so the time cannot change.
Yes, because the interval proves training causes faster times for all runners.
Yes, because the entire interval is negative and corresponds to a decrease between 0.12 and 0.28 minutes per week.
No, because the interval includes -0.20, which means the slope is exactly -0.20.
Explanation
This question tests evaluating range claims for regression slopes using confidence intervals in AP Statistics. The scientist claims training reduces 5K time by 0.10 to 0.30 minutes per week, meaning slope between -0.30 and -0.10. The 95% confidence interval (-0.28, -0.12) corresponds to decreases of 0.12 to 0.28, entirely within 0.10 to 0.30, supporting the claim. Choice E distracts by misinterpreting inclusion of -0.20 as exact equality, but intervals estimate ranges. Mini-lesson: for a slope range claim, check if the entire interval falls within the claimed range; if yes, it's supported; if it extends outside, not. For example, (-0.25, -0.15) within (-0.30, -0.10) supports, but (-0.35, -0.05) does not.
A nutrition researcher fits a regression model to predict LDL cholesterol (mg/dL) from weekly servings of oats. The researcher claims that eating more oats reduces LDL cholesterol. A 99% confidence interval for the slope is $(-4.1, 0.3)$ mg/dL per serving. Is the claim supported by the confidence interval?
No, because the interval includes 0, so a slope of 0 is plausible and the claim is not supported.
No, because a 99% interval is too wide to interpret.
Yes, because most of the interval is negative, so the slope must be negative.
No, because negative values in the interval mean the regression line is curved.
Yes, because the interval includes 0, which proves oats reduce LDL.
Explanation
This question focuses on justifying regression slope claims with confidence intervals in AP Statistics. The researcher claims more oats reduce LDL cholesterol, implying a negative slope. The 99% confidence interval (-4.1, 0.3) includes 0 and positive values, meaning no effect or an increase is plausible, so the claim isn't supported. Choice A distracts by saying most of the interval is negative guarantees negativity, but the inclusion of non-negative values refutes this. Mini-lesson: to support a directional claim like negative slope, the interval must be entirely negative; inclusion of 0 or positives means the claim lacks evidence. For example, (-3, -1) supports negativity, but (-3, 1) does not.
A manager regresses weekly sales ($y$, dollars) on number of employees scheduled ($x$). The manager claims, “Scheduling more employees increases mean weekly sales.” A 98% confidence interval for the slope is $( -55,\ -10)$ dollars per employee. Is the claim supported by the confidence interval?
No, because the entire interval is below 0, which supports a negative association (mean sales decrease as employees increase), contradicting the claim.
No, because a confidence interval for slope only applies if the relationship is causal, which is not stated.
Yes, because negative slope means sales go up when employees go up.
Yes, and it proves adding employees will cause sales to rise by $10$ to $55$ dollars each week.
Yes, because the interval does not include 0, so scheduling more employees increases sales.
Explanation
The manager claims that scheduling more employees increases mean weekly sales - a positive association. However, the 98% confidence interval (-55, -10) contains only negative values. This means we're 98% confident the true slope is negative, indicating that as employees increase, mean sales actually decrease. Choice B correctly identifies that this contradicts the claim. The entirely negative interval supports a negative association, which is the opposite of what the manager claimed. This is a clear case where the data contradicts the proposed relationship.
An economist regresses monthly household savings ($y$, dollars) on monthly income ($x$, dollars). The economist claims, “Higher income is associated with higher mean savings.” A 90% confidence interval for the slope is $(0.05,\ 0.22)$ dollars saved per dollar of income. Is the claim supported by the confidence interval?
Yes, because the entire interval is above 0, supporting a positive association between income and mean savings.
Yes, because 90% of households save between 0.05 and 0.22 dollars each month.
No, because a confidence interval for slope must include 0 to indicate statistical significance.
Yes, and it proves that increasing income will cause savings to increase for all households.
No, because the interval is not symmetric around its midpoint, so it cannot support any claim.
Explanation
The economist claims higher income is associated with higher mean savings - a positive association. The 90% confidence interval (0.05, 0.22) contains only positive values and doesn't include 0. This means we're 90% confident the true slope is positive, supporting the claim. Choice B correctly identifies this. The interval tells us that for each additional dollar of income, mean savings increase by between $0.05 and $0.22. When an entire confidence interval is above 0, it provides evidence for a positive association between the variables at the stated confidence level.
A teacher models course grade (percent) from number of absences for a random sample of students. The teacher claims: “More absences lead to lower mean course grades.” A 90% confidence interval for the slope is $(-2.5,\ 0.3)$ percent per absence. Is the claim supported by the confidence interval?
No, because the interval includes 0, which proves there is no relationship at all.
Yes, because 90% of students’ grades will drop by between 0.3 and 2.5 points for each additional absence.
No, because the interval includes 0, so a slope of 0 is plausible and the data do not provide convincing evidence of a negative association.
Yes, because the confidence interval shows absences cause grades to decrease.
Yes, because the interval includes negative values, so the slope must be negative.
Explanation
This question tests understanding of confidence intervals that include both positive and negative values. The claim states 'more absences lead to lower mean course grades,' which requires a negative slope. The 90% confidence interval (-2.5, 0.3) includes both negative and positive values, and importantly, it includes 0. When a confidence interval for slope contains 0, it means a slope of 0 is plausible, indicating we don't have convincing evidence of either a positive or negative association at the given confidence level. Choice B correctly recognizes that including 0 means the data do not provide convincing evidence of the claimed negative association. Even though most of the interval is negative, the inclusion of positive values and 0 prevents us from concluding the slope is definitely negative.
A nutrition researcher models systolic blood pressure ($y$) using daily sodium intake ($x$, mg). The researcher claims, “Greater sodium intake is associated with higher mean systolic blood pressure.” A 95% confidence interval for the slope is $(-0.006,\ 0.014)$ mmHg per mg. Is the claim supported by the confidence interval?
Yes, because the interval contains positive values, so the association must be positive.
Yes, because 95% of people’s blood pressures change by between -0.006 and 0.014 mmHg per mg.
No, because the interval includes 0, so a slope of 0 is plausible and the claim is not supported by the interval.
Yes, and it proves sodium intake causes blood pressure to increase.
No, because the slope must be negative whenever the interval includes a negative number.
Explanation
The researcher claims greater sodium intake is associated with higher mean blood pressure - a positive association. The 95% confidence interval (-0.006, 0.014) includes both negative and positive values, and importantly includes 0. When 0 is in the interval, we cannot conclude there's a clear association in either direction. Choice B correctly identifies that the claim is not supported. At the 95% confidence level, we cannot rule out that there's no association (slope = 0) between sodium intake and blood pressure, so the claim of a positive association is not supported by this interval.
A teacher models exam score ($y$) using number of practice problems completed ($x$). The teacher claims, “Completing more practice problems is associated with higher mean exam scores.” A 95% confidence interval for the slope is $(1.4,\ 3.6)$ points per problem. Is the claim supported by the confidence interval?
No, because confidence intervals only describe correlation, which is unrelated to slope.
No, because the interval does not include 0, so the slope cannot be positive.
Yes, because the entire interval is above 0, supporting a positive slope (positive association).
Yes, and it guarantees every student’s score will increase by 1.4 to 3.6 points for each additional problem.
Yes, because 95% of exam scores will fall between 1.4 and 3.6 points.
Explanation
The teacher claims that more practice problems are associated with higher mean exam scores - a positive association. The 95% confidence interval (1.4, 3.6) contains only positive values and doesn't include 0. This means we're 95% confident the true slope is positive, supporting the claim. Choice B correctly identifies this. The interval tells us about the average change in exam scores per additional practice problem, not guarantees for individual students. When an entire interval is above 0, it provides strong evidence for a positive association between the variables.
A real estate analyst fits a regression of home sale price ($y$, in thousands of dollars) on home size ($x$, in square feet). The analyst claims, “Each additional square foot increases the mean sale price.” A 99% confidence interval for the slope is $( -0.002,\ 0.090)$ thousand dollars per square foot. Is the claim supported by the confidence interval?
Yes, and it proves that increasing square footage causes price to rise for all homes.
No, because a 99% confidence interval is too conservative to use for claims about slope.
Yes, because 99% of homes will have price changes between -0.002 and 0.090 thousand dollars per square foot.
No, because the interval contains 0, so a slope of 0 is plausible and the claim is not clearly supported.
Yes, because the upper endpoint is positive, so the slope must be positive.
Explanation
The analyst claims that each additional square foot increases mean sale price - a positive association. The 99% confidence interval (-0.002, 0.090) includes both negative and positive values, and importantly includes 0. When 0 is in the confidence interval, we cannot conclude there's a clear positive or negative association at the given confidence level. Choice B correctly identifies that the claim is not clearly supported. A slope of 0 (no association) is plausible within our 99% confidence interval, so we cannot support the claim of a positive association.
A city planner regresses average commute time ($y$, minutes) on distance from downtown ($x$, miles). The planner claims, “Living farther from downtown is associated with longer mean commute times.” A 90% confidence interval for the slope is $(0.8,\ 2.1)$ minutes per mile. Is the claim supported by the confidence interval?
Yes, because the entire interval is above 0, supporting the claim of a positive association.
Yes, and it proves that moving farther away will cause each person’s commute to increase by 0.8 to 2.1 minutes per mile.
No, because the interval should include 0 to show a real relationship.
No, because 0.8 to 2.1 is too wide to support any claim.
No, because a confidence interval for slope cannot indicate direction (positive or negative).
Explanation
The city planner claims that living farther from downtown is associated with longer mean commute times - a positive association. The 90% confidence interval (0.8, 2.1) contains only positive values and doesn't include 0. This supports the claim of a positive association at the 90% confidence level. Choice A correctly identifies this. The interval estimates the average increase in commute time per additional mile from downtown. Confidence intervals for slope absolutely can indicate direction when they don't contain 0, and the width of the interval doesn't invalidate the directional conclusion.
A school counselor fits a least-squares regression of students’ semester GPA ($y$) on average hours of sleep per night ($x$). The counselor claims, “Each additional hour of sleep is associated with an increase in mean GPA.” A 95% confidence interval for the slope is $(0.03,\ 0.18)$ GPA points per hour. Is the claim supported by the confidence interval?
No, because the interval is not centered at 0, so the slope could still be 0.
Yes, because 95% of individual students’ slopes must fall between 0.03 and 0.18.
Yes, because the entire interval is above 0, supporting a positive association between $x$ and the mean of $y$.
Yes, and it proves that increasing sleep will cause GPA to increase for every student.
No, because a confidence interval cannot be used to evaluate a claim about a slope.
Explanation
This question tests whether you can use a confidence interval for slope to evaluate a claim about association. The counselor claims a positive association between sleep hours and mean GPA. The 95% confidence interval (0.03, 0.18) contains only positive values - it doesn't include 0. This means we're 95% confident the true slope is positive, supporting the claim of a positive association. Choice B correctly identifies this. The interval tells us about the slope of the regression line (which describes the mean response), not individual student outcomes, making choices D and E incorrect.