AP Statistics Quiz: Justifying Claims Slope Of Regression Models

What this quiz covers

This quiz focuses on Justifying Claims Slope Of Regression Models, 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.

Question 1

1. Yes, because the entire interval is negative and corresponds to a decrease between 0.12 and 0.28 minutes per week. (correct answer)
2. No, because the interval is negative, so it contradicts any claim about reduction.
3. No, because the interval does not include 0, so the time cannot change.
4. Yes, because the interval proves training causes faster times for all runners.
5. 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.

Question 2

1. Yes, because most of the interval is negative, so the slope must be negative.
2. Yes, because the interval includes 0, which proves oats reduce LDL.
3. No, because the interval includes 0, so a slope of 0 is plausible and the claim is not supported. (correct answer)
4. No, because a 99% interval is too wide to interpret.
5. No, because negative values in the interval mean the regression line is curved.

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.

Question 3

1. Yes, because the interval does not include 0, so scheduling more employees increases sales.
2. No, because the entire interval is below 0, which supports a negative association (mean sales decrease as employees increase), contradicting the claim. (correct answer)
3. Yes, because negative slope means sales go up when employees go up.
4. No, because a confidence interval for slope only applies if the relationship is causal, which is not stated.
5. Yes, and it proves adding employees will cause sales to rise by $10$ to $55$ dollars each week.

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.

Question 4

1. No, because the interval is not symmetric around its midpoint, so it cannot support any claim.
2. Yes, because the entire interval is above 0, supporting a positive association between income and mean savings. (correct answer)
3. Yes, and it proves that increasing income will cause savings to increase for all households.
4. No, because a confidence interval for slope must include 0 to indicate statistical significance.
5. Yes, because 90% of households save between 0.05 and 0.22 dollars each month.

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.

Question 5

1. Yes, because the interval includes negative values, so the slope must be negative.
2. 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. (correct answer)
3. Yes, because 90% of students' grades will drop by between 0.3 and 2.5 points for each additional absence.
4. No, because the interval includes 0, which proves there is no relationship at all.
5. Yes, because the confidence interval shows absences cause grades to decrease.

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.

Question 6

1. Yes, because the interval contains positive values, so the association must be positive.
2. No, because the interval includes 0, so a slope of 0 is plausible and the claim is not supported by the interval. (correct answer)
3. Yes, and it proves sodium intake causes blood pressure to increase.
4. No, because the slope must be negative whenever the interval includes a negative number.
5. Yes, because 95% of people's blood pressures change by between -0.006 and 0.014 mmHg per mg.

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.

Question 7

1. No, because the interval does not include 0, so the slope cannot be positive.
2. Yes, because the entire interval is above 0, supporting a positive slope (positive association). (correct answer)
3. Yes, and it guarantees every student's score will increase by 1.4 to 3.6 points for each additional problem.
4. No, because confidence intervals only describe correlation, which is unrelated to slope.
5. 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.

Question 8

1. Yes, because the upper endpoint is positive, so the slope must be positive.
2. No, because the interval contains 0, so a slope of 0 is plausible and the claim is not clearly supported. (correct answer)
3. Yes, and it proves that increasing square footage causes price to rise for all homes.
4. No, because a 99% confidence interval is too conservative to use for claims about slope.
5. Yes, because 99% of homes will have price changes between -0.002 and 0.090 thousand dollars per square foot.

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.

Question 9

1. Yes, because the entire interval is above 0, supporting the claim of a positive association. (correct answer)
2. No, because a confidence interval for slope cannot indicate direction (positive or negative).
3. No, because 0.8 to 2.1 is too wide to support any claim.
4. 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.
5. No, because the interval should include 0 to show a real relationship.

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.

Question 10

1. No, because the interval is not centered at 0, so the slope could still be 0.
2. Yes, because the entire interval is above 0, supporting a positive association between $x$ and the mean of $y$. (correct answer)
3. No, because a confidence interval cannot be used to evaluate a claim about a slope.
4. Yes, and it proves that increasing sleep will cause GPA to increase for every student.
5. Yes, because 95% of individual students' slopes must fall between 0.03 and 0.18.

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.

Question 11

1. No, because the interval does not contain 0, so the slope could still be negative.
2. Yes, because the entire interval is above 0, so the data support a positive slope (an increase in mean score per hour). (correct answer)
3. No, because a confidence interval cannot be used to assess whether a slope is positive or negative.
4. Yes, because the interval proves that tutoring causes higher exam scores.
5. Yes, because 95% of individual students will increase their score by between 1.2 and 3.8 points for each additional hour.

Explanation: This question tests whether you can use a confidence interval for slope to evaluate a claim about a positive association. The claim states that 'each additional hour of tutoring increases the mean final exam score,' which is a claim about a positive slope. The 95% confidence interval (1.2, 3.8) contains only positive values and does not include 0. When a confidence interval for slope contains only positive values, we have convincing evidence that the true slope is positive, supporting the claim of a positive association. Choice B correctly identifies this reasoning. Remember: if the interval doesn't contain 0 and all values have the same sign, we have evidence for that direction of association.

Question 12

1. Yes, because the slope must be positive since the upper end of the interval is positive.
2. No, because the interval includes 0, so a slope of 0 is plausible and the data do not provide convincing evidence of a positive association. (correct answer)
3. Yes, because including 0 means the slope is exactly 0.075.
4. Yes, because the interval proves that increasing square footage will cause the price to increase.
5. No, because confidence intervals only apply to correlation, not to regression slopes.

Explanation: This question tests understanding of confidence intervals that include 0 as an endpoint. The claim states 'larger homes tend to sell for more, on average,' which requires a positive slope. The 95% confidence interval (0.0, 0.15) includes 0 as its lower endpoint. When 0 is included in a confidence interval for slope, even as an endpoint, it means a slope of 0 is plausible. This indicates we don't have convincing evidence of 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 positive association. Remember: any value within the interval, including endpoints, represents a plausible value for the true slope.

Question 13

1. Yes, because the interval contains positive values, so the slope must be positive.
2. No, because the interval includes 0, so a zero slope is plausible and the data do not clearly support a positive association. (correct answer)
3. Yes, because the interval includes 0, which means the relationship is definitely positive but weak.
4. No, because the interval includes negative values, which proves temperature lowers ozone.
5. Yes, because 90% of days will have ozone increase by between -0.4 and 2.1 ppb for every 1°F increase.

Explanation: This question asks whether a confidence interval supports a claim about positive association between temperature and ozone levels. The claim states 'higher temperatures are associated with higher mean ozone levels,' which requires a positive slope. The 90% confidence interval (-0.4, 2.1) 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. Choice B correctly identifies that because 0 is included, the data do not clearly support the claimed positive association. A confidence interval that includes 0 suggests the relationship could be positive, negative, or nonexistent.

Question 14

1. Yes, because the interval does not include 0, so advertising definitely increases sales.
2. No, because the entire interval is below 0, which supports a negative slope rather than an increase. (correct answer)
3. Yes, because 99% confidence guarantees the slope is positive.
4. No, because a confidence interval cannot be used to make any claim about direction.
5. Yes, because the interval means 99% of months will have sales decrease by between 0.2 and 1.8 thousand units when advertising increases.

Explanation: This question tests whether students can identify when a confidence interval contradicts a claim. The claim states 'increasing advertising spending increases mean monthly sales,' which requires a positive slope. However, the 99% confidence interval (-1.8, -0.2) contains only negative values. When a confidence interval contains only negative values, it provides convincing evidence of a negative slope, meaning as advertising increases, mean sales decrease. Choice B correctly identifies that the entirely negative interval contradicts the claim of increased sales. This illustrates an important point: when evaluating claims with confidence intervals, check whether the interval's values match the direction claimed (positive for increases, negative for decreases).

Question 15

1. No, because the interval is close to 0, so the slope must be 0.
2. Yes, because the entire interval is above 0, supporting a positive slope (higher sodium, higher mean blood pressure). (correct answer)
3. No, because the confidence interval refers to individual blood pressures, not the mean response.
4. Yes, because the interval proves sodium intake causes blood pressure to rise.
5. No, because a positive slope would require the interval to include 0.

Explanation: This question asks whether a confidence interval supports a claim about positive association between sodium intake and blood pressure. The claim states 'higher sodium intake is associated with higher mean systolic blood pressure,' which requires a positive slope. The 95% confidence interval (0.002, 0.009) contains only positive values and does not include 0. When a confidence interval for slope contains only positive values, we have convincing evidence that the true slope is positive, supporting the claim of a positive association. Choice B correctly identifies this reasoning. Note that the small magnitude of the values doesn't affect the interpretation—what matters is that all values are positive and 0 is not included.

Question 16

1. No, because the interval has zero width, so it cannot be used to support any claim.
2. Yes, because the entire interval is above 0, supporting a positive slope (even if estimated very precisely). (correct answer)
3. No, because a confidence interval must include 0 to show a positive slope.
4. Yes, because the interval proves income causes people to save more.
5. Yes, because 95% of households increase their savings rate by exactly 0.10 when income rises by \1{,}000.

Explanation: This question presents an unusual confidence interval where both endpoints are the same: (0.10, 0.10). This indicates an extremely precise estimate of the slope, likely due to a very large sample size or very little variability. The claim states 'higher income is associated with a higher mean savings rate,' which requires a positive slope. Since the interval contains only the positive value 0.10 and does not include 0, we have convincing evidence of a positive slope. Choice B correctly identifies that even with this precise interval, the reasoning remains the same: all values are positive and 0 is not included, so the claim is supported. The precision of the estimate doesn't change the interpretation method.

Question 17

1. Yes, because 0 is not in the interval, so the slope could still be 0.
2. No, because the entire interval is above 0, which contradicts a slope of 0. (correct answer)
3. Yes, because the interval indicates the slope is exactly 0.50 species per hectare.
4. No, because confidence intervals are only about individual predictions, not the slope.
5. Yes, because the interval proves that increasing park area causes species richness to increase.

Explanation: In AP Statistics, this question evaluates justifying claims about the slope using confidence intervals. The ecologist claims no linear association between park area and mean species richness, meaning a slope of 0. The 95% confidence interval (0.12, 0.88) is entirely above 0, contradicting the claim as 0 is not plausible. Choice A is a distractor because although 0 is not in the interval, this rejects the claim of slope=0 rather than supporting it. Choice E incorrectly implies causation, but the interval only estimates the population slope without proving cause. Mini-lesson: To support a claim of no association (slope=0), the interval must include 0; if it excludes 0, the data suggest a non-zero slope at that confidence level.

Question 18

1. Yes, because most of the interval is negative, so the slope must be negative.
2. No, because the interval includes $0$, so a slope of $0$ is plausible. (correct answer)
3. Yes, because the interval includes negative values, so the slope is definitely negative.
4. No, because a confidence interval cannot indicate the direction of an association.
5. Yes, and this proves increasing mileage causes faster 5K times.

Explanation: This question asks whether a confidence interval supports a claim of a negative slope. The 98% confidence interval (-0.40, 0.05) includes both negative and positive values, and importantly includes 0. Since 0 is a plausible value, we cannot conclude the slope is negative. Choice A incorrectly focuses on "most" of the interval being negative rather than considering all plausible values. When a confidence interval contains 0, it indicates that no linear relationship (slope = 0) is plausible, so we cannot support claims about the direction of the relationship.

Question 19

1. Yes, because the upper endpoint is positive, so the slope must be positive.
2. No, because the interval includes 0, so the data do not provide convincing evidence of a positive slope. (correct answer)
3. Yes, because 99% confidence means the slope is positive with 99% probability.
4. No, because the negative endpoint proves sodium decreases blood pressure for everyone.
5. Yes, because the interval is narrow, so the relationship is strong and positive.

Explanation: This question examines whether a confidence interval supports a claim about a positive relationship between sodium intake and blood pressure. The 99% confidence interval (-0.004, 0.012) includes both negative and positive values, and crucially, it contains 0. When 0 is in the confidence interval for slope, we cannot conclude that there is a linear relationship in either direction - the true slope might be zero (no relationship). Therefore, the data do not provide convincing evidence of a positive slope as claimed. Choice B correctly identifies this. The fact that we used 99% confidence (rather than 95% or 90%) doesn't change the interpretation - if 0 is in the interval, we lack evidence for the claimed direction. This is a key principle in using confidence intervals for hypothesis testing.

Question 20

1. Yes, because 0 is in the interval, so a slope of 0 is plausible at the 90% level. (correct answer)
2. No, because the interval contains negative values, so the slope must be negative.
3. No, because 0 is in the interval, which proves the slope is exactly 0.
4. Yes, because the interval guarantees ozone will not change when traffic changes.
5. No, because the interval shows traffic causes ozone to change, so the slope cannot be 0.

Explanation: This AP Statistics question tests evaluating claims of no association using slope confidence intervals. The scientist claims no linear relationship between traffic and mean ozone level, meaning slope=0. The 90% confidence interval (-0.006, 0.002) includes 0, making slope=0 plausible and supporting the claim. Choice C distracts by saying inclusion of 0 proves exactly 0, but it only shows it's possible. Choice E assumes causation in the wrong direction. Mini-lesson: Claims of slope=0 are supported if 0 is within the interval; exclusion of 0 would indicate evidence against no association.