- A real estate analyst fits a regression model predicting selling price (in thousands of dollars) from square footage for a random sample of homes in a city. The analyst claims: "Larger homes tend to sell for more, on average." A 95% confidence interval for the slope is (0.0, 0.15) thousand dollars per square foot. Is the claim supported by the confidence interval?
Answer: 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.
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.
- A city planner fits a regression model predicting average commute time (y) from distance from downtown in miles (x). The planner claims that commute time increases by at least 1.5 minutes per mile (slope ≥1.5). A 90% confidence interval for the slope is (1.6, 2.4) minutes per mile. Is the claim supported by the confidence interval?
Answer: Yes, because the entire interval is at or above 1.5, so all plausible slopes satisfy the claim.
This question asks whether a confidence interval supports a claim about a minimum slope value. The planner claims slope ≥ 1.5, and the 90% confidence interval (1.6, 2.4) contains only values at or above 1.5. Since all plausible slopes satisfy the claim (they're all at least 1.5), the claim is supported. Choice A incorrectly states that confidence intervals cannot support directional claims, when in fact they are commonly used for this purpose. When the entire confidence interval satisfies the inequality in the claim, the claim is supported by the data.
- 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?
Answer: No, because the interval includes 0, so a slope of 0 is plausible and the claim is not supported.
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 real estate analyst uses regression to predict house price (in $1000s) from square footage. The analyst claims that each additional square foot increases the predicted price. A 95% confidence interval for the slope is (−0.02,0.10) thousand dollars per square foot. Is the claim supported by the confidence interval?
Answer: No, because the interval includes 0 and negative values, so a positive slope is not clearly supported.
This question assesses justifying positive slope claims with confidence intervals in AP Statistics. The analyst claims each square foot increases house price, implying a positive slope. The 95% confidence interval (−0.02,0.10) includes negative values and 0, so no increase or a decrease is plausible, not supporting the claim. Choice A distracts by saying positive values force a positive slope, ignoring negatives. Mini-lesson: for a positive slope claim, the interval must be entirely >0; inclusion of ≤0 means the claim lacks support. For example, (0.05,0.15) supports an increase, but (−0.05,0.15) does not. - A city analyst models monthly electricity use (kWh) from average monthly temperature (°F) using linear regression. The analyst claims that higher temperature is associated with higher mean electricity use. A 90% confidence interval for the slope is (−3.2,1.5) kWh per °F. Is the claim supported by the confidence interval?
Answer: No, because the interval includes 0, so the data do not provide convincing evidence that the slope is positive.
This question asks whether a confidence interval supports a claim about a positive slope (higher temperature associated with higher electricity use). The 90% confidence interval (-3.2, 1.5) includes both negative and positive values, and importantly, it contains 0. When a confidence interval for slope contains 0, it means we cannot rule out the possibility that there is no linear relationship (slope = 0). Therefore, the data do not provide convincing evidence that the slope is positive as claimed. Choice B correctly identifies this key principle. The interval suggests the true slope could be negative, zero, or positive - we simply don't have enough evidence to conclude it's positive. This is a fundamental concept in hypothesis testing using confidence intervals.
- A teacher fits a regression model predicting exam score (points) from the number of absences during the semester. The teacher claims that more absences are associated with a lower mean exam score. A 98% confidence interval for the slope is (−2.4,−0.1) points per absence. Is the claim supported by the confidence interval?
Answer: Yes, because the entire interval is negative, supporting a negative slope (lower mean score with more absences).
This question examines whether a confidence interval supports a claim about absences being associated with lower exam scores (negative slope). The 98% confidence interval (-2.4, -0.1) contains only negative values, meaning we are 98% confident the true slope is negative. This supports the teacher's claim that more absences are associated with lower mean exam scores. Choice A correctly identifies this - when an entire interval is negative, it provides evidence for a negative relationship. The confidence level (98% vs 95% or 90%) doesn't affect how we interpret the interval; it only affects how confident we are. The interval gives a range of plausible values for the slope, not a single exact value, and it describes association, not causation for every student.
- A nutrition researcher uses linear regression to predict resting heart rate (y) from daily caffeine intake in mg (x). The researcher claims the slope is 0 (no linear relationship). A 95% confidence interval for the slope is (−0.03, 0.11) beats per minute per mg. Is the claim supported by the confidence interval?
Answer: Yes, because 0 is contained in the interval, so a slope of 0 is plausible.
This question tests understanding of confidence intervals that contain zero. The 95% confidence interval (-0.03, 0.11) includes 0, meaning a slope of 0 (no linear relationship) is a plausible value. This supports the researcher's claim that the slope equals 0. Choices B and C incorrectly conclude the slope must be positive or negative based on the interval containing those values, but when 0 is included, we cannot rule out no relationship. When evaluating whether a slope equals 0, check if 0 falls within the confidence interval; if it does, the claim is supported. - A real estate agent models house price (in $1000s) from distance to downtown (miles). The agent claims that being farther from downtown is associated with a lower mean house price. A 95% confidence interval for the slope is (−12.5,−3.0) (thousand dollars per mile). Is the claim supported by the confidence interval?
Answer: Yes, because the entire interval is negative, supporting a negative slope (lower mean price as distance increases).
This question tests interpretation of a confidence interval for a claim about a negative slope. The agent claims that greater distance from downtown is associated with lower mean house price, which requires a negative slope. The 95% confidence interval (-12.5, -3.0) contains only negative values, meaning we are 95% confident the true slope is negative (between -$12,500 and -$3,000 per mile). This supports the claim of a negative association. Choice A correctly identifies this - when an entire interval is negative, it provides evidence for a negative relationship. The interval describes the association in the population, not a causal effect for every individual house. Confidence intervals for slopes work the same way regardless of whether predictors are quantitative or categorical.
- A city planner fits a regression model predicting public transit ridership (thousands of rides per day) from gasoline price (dollars per gallon) using data from randomly selected days. The planner claims: "Higher gas prices are associated with higher mean transit ridership." A 95% confidence interval for the slope is (0.8, 5.4) thousand rides per $$1 increase in gas price. Is the claim supported by the confidence interval?
Answer: Yes, because the entire interval is above 0, supporting a positive slope (higher gas price, higher mean ridership).
This question examines whether a confidence interval supports a claim about positive association between gas prices and transit ridership. The claim states 'higher gas prices are associated with higher mean transit ridership,' which requires a positive slope. The 95% confidence interval (0.8, 5.4) 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 while the interval supports the association claim, it doesn't prove causation—it only indicates that in the data, higher gas prices tend to occur with higher ridership.
- A sports scientist uses a regression model to predict 100-meter time (seconds) from weekly sprint-training hours for a random sample of competitive runners. The scientist claims: "More sprint training is associated with lower mean 100-meter times." A 95% confidence interval for the slope is (−0.30, −0.05) seconds per hour. Is the claim supported by the confidence interval?
Answer: Yes, because the entire interval is below 0, supporting a negative slope (more training, lower mean time).
This question examines whether a confidence interval supports a claim about negative association. The claim states 'more sprint training is associated with lower mean 100-meter times,' which requires a negative slope (as time decreases when training increases). The 95% confidence interval (-0.30, -0.05) contains only negative values and does not include 0. When a confidence interval for slope contains only negative values, we have convincing evidence that the true slope is negative, supporting the claim of a negative association. Choice A correctly identifies this reasoning. Note that negative slopes indicate that as the explanatory variable increases, the mean response variable decreases, which matches the claim.
- A botanist models plant height (y, cm) using amount of fertilizer applied (x, grams). The botanist claims, "More fertilizer is associated with taller plants on average." A 95% confidence interval for the slope is (0.00, 0.42) cm per gram. Is the claim supported by the confidence interval?
Answer: No, because the interval includes 0, so a slope of 0 is plausible and the claim of a positive association is not clearly supported.
The botanist claims more fertilizer is associated with taller plants on average - a positive association. The 95% confidence interval (0.00, 0.42) technically includes 0 at its lower endpoint. When a confidence interval includes 0 (even as an endpoint), we cannot conclusively support a claim of positive or negative association at that confidence level. Choice B correctly identifies this. A slope of exactly 0 is within the plausible range of values, so we cannot clearly support the claim of a positive association. The interval suggests the association could be anywhere from no effect to a moderate positive effect.
- 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?
Answer: Yes, because the entire interval is above 0, supporting the claim of a positive association.
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.
- An environmental scientist models ozone level (y) from traffic volume (x) on 30 days. The scientist claims that traffic volume has no linear relationship with mean ozone level. A 90% confidence interval for the slope is (−0.006,0.002) (ozone units per traffic unit). Is the claim supported by the confidence interval?
Answer: Yes, because 0 is in the interval, so a slope of 0 is plausible at the 90% level.
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.
- A real estate analyst fits a regression model predicting house sale price (y) from square footage (x). The analyst claims that each additional square foot increases mean sale price by more than 150 dollars (slope >150). A 99% confidence interval for the slope is (120, 190) dollars per square foot. Is the claim supported by the confidence interval?
Answer: No, because the interval includes values at or below 150, so slope >150 is not guaranteed.
This question asks whether a confidence interval supports a claim that the slope exceeds a specific value. The 99% confidence interval (120, 190) contains all plausible slope values. The analyst claims slope > 150, but the interval includes values at or below 150 (like 120, 130, 140, 150), so we cannot be confident the true slope exceeds 150. Choice A incorrectly focuses on the upper bound rather than considering all values in the interval. To support a claim that a parameter is greater than some value, the entire confidence interval must be above that value. - A college investigates whether attending review sessions (x, number of sessions) predicts final course grade (y) for 55 students. The professor claims that each additional session increases the mean grade by more than 2 points. A 95% confidence interval for the slope is (1.1,2.6) grade points per session. Is the claim supported by the confidence interval?
Answer: No, because the interval includes values at or below 2, so "more than 2" is not supported by the CI.
AP Statistics questions like this evaluate using confidence intervals for 'more than' claims about slopes. The professor claims each additional session increases mean grade by more than 2 points, so slope >2. The 95% confidence interval (1.1, 2.6) includes values ≤2, not supporting the claim. Choice A is a distractor because while it includes >2, the lower end allows ≤2 as plausible. Choice E confuses association with causation. Mini-lesson: For a claim of slope >k, the entire interval must exceed k; if it includes or goes below k, the claim is not supported at that level.
- An environmental scientist models daily ozone level (y) using daily high temperature (x) in a city. The scientist claims there is a negative linear association, meaning the true slope is less than 0. A 90% confidence interval for the slope is (−1.8, −0.4) (ozone units per degree). Is the claim supported by the confidence interval?
Answer: Yes, because all plausible slope values in the interval are less than 0.
This question asks whether a confidence interval supports a claim about a negative slope. The 90% confidence interval (-1.8, -0.4) contains only negative values, meaning all plausible slopes are less than 0. This fully supports the scientist's claim that the true slope is negative (less than 0). Choice B incorrectly states that not including 0 means the slope could be either positive or negative, when in fact the entire interval being negative confirms a negative relationship. When a confidence interval is entirely on one side of a value, it provides strong evidence that the true parameter is on that same side. - An ecologist models the number of bird species observed at a site from the size of the site (acres). The ecologist claims that larger sites are associated with more species on average. A 95% confidence interval for the slope is (0.02,0.11) species per acre. Is the claim supported by the confidence interval?
Answer: Yes, because the entire interval is positive, supporting a positive slope (more species as acres increase).
This question tests understanding of using confidence intervals to evaluate claims about positive relationships. The ecologist claims larger sites have more bird species on average, which requires a positive slope. The 95% confidence interval (0.02, 0.11) contains only positive values, meaning we are 95% confident the true slope is positive (between 0.02 and 0.11 species per acre). This directly supports the claim. Choice B correctly identifies this - when the entire interval is positive, it provides evidence for a positive association. The interval describes the average relationship across sites, not a guarantee about what happens at every individual site. A confidence interval that doesn't include 0 actually indicates a statistically significant slope, contrary to what choice E suggests.
- A manager models weekly sales (in thousands of dollars) from the number of online ads purchased. The manager claims that buying more ads has no linear effect on weekly sales. A 95% confidence interval for the slope is (−0.8,1.1) thousand dollars per ad. Is the claim supported by the confidence interval?
Answer: Yes, because 0 is in the interval, so a slope of 0 is plausible.
This question tests justifying claims about no effect (slope=0) in regression models using confidence intervals in AP Statistics. The manager claims more ads have no linear effect on sales, meaning the slope is 0. The 95% confidence interval (-0.8, 1.1) includes 0, so a slope of 0 is plausible, supporting the claim. Choice E is a distractor, incorrectly stating intervals can't address effects, but they directly test if 0 is plausible. Mini-lesson on slope claims: for 'no effect,' check if 0 is in the interval; if yes, it's supported; if the interval excludes 0, no effect is implausible. For instance, (0.1, 0.5) rejects no effect, but (-0.2, 0.3) supports it.
- A marketing team models weekly sales (y) from weekly advertising spending in thousands of dollars (x) using 25 weeks. The team claims that spending has a positive linear effect on mean sales. A 95% confidence interval for the slope is (−3.2,14.5) (sales units per $$1{,}000). Is the claim supported by the confidence interval?
Answer: No, because the interval includes 0, so a positive slope is not supported at the 95% level.
This question in AP Statistics examines justifying slope claims with confidence intervals. The team claims a positive linear effect of advertising on mean sales, requiring a slope greater than 0. The 95% confidence interval (-3.2, 14.5) includes negative values and 0, so it does not support a positive slope. Choice A distracts by focusing only on the positive upper bound, ignoring the negative possibilities. Choice E assumes causation, but intervals do not establish cause, only plausible slope values. Mini-lesson: For a positive slope claim, the interval must be entirely positive; inclusion of 0 or negatives means the association could be zero or opposite at the given confidence level.
- A fitness researcher uses a least-squares line to predict resting heart rate (y) from weekly minutes of aerobic exercise (x) for 80 adults. The researcher claims that more exercise is associated with a lower mean resting heart rate. A 99% confidence interval for the slope is (−0.09,−0.01) beats per minute per minute of exercise. Is the claim supported by the confidence interval?
Answer: Yes, because the entire interval is below 0, supporting a negative slope (lower mean heart rate with more exercise).
This AP Statistics question focuses on using confidence intervals to justify claims about regression slopes. The researcher claims more exercise is associated with lower mean resting heart rate, indicating a negative slope. The 99% confidence interval (-0.09, -0.01) is entirely below 0, supporting the claim. Choice D is a distractor as it misapplies the interval to individuals instead of the mean heart rate. Choice E errs by claiming proof of causation, whereas the interval only provides evidence of association. Mini-lesson: A claim of negative slope is supported if the confidence interval lies wholly below 0; higher confidence levels like 99% make intervals wider, but here it still confirms the direction.
