AP Statistics Quiz: Confidence Intervals Slope Of Regression Models

What this quiz covers

This quiz focuses on Confidence Intervals 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. We are 95% confident that for each additional 100 square feet, the mean selling price in the population increases by between \8{,}000 and \14{,}000. (correct answer)
2. There is a 95% probability that the selling price of a randomly selected house is between \8{,}000 and \14{,}000.
3. Because the interval does not include 0, the correlation between size and price is between 8 and 14.
4. We are 95% confident that 95% of houses increase in price by between \8{,}000 and \14{,}000 when size increases by 100 square feet.
5. We are 95% confident that the slope of the sample regression line will be between 8 and 14 for these same 30 houses if we recompute it.

Explanation: This question assesses interpreting a 95% confidence interval for the slope β in regressing house price on size. The interval (8, 14) implies we are 95% confident that β, the average increase in mean price per 100 square feet, is between $8,000 and $14,000. Choice C distracts by equating the interval to correlation, but correlation is not measured in the same units or scale. Choice D wrongly extends the interval to percentages of houses rather than the population mean. Mini-lesson: Slope confidence intervals reflect the range where the true population rate of change likely falls, incorporating sampling error; positive endpoints indicate an upward trend, and proper unit interpretation is key, as slopes depend on variable scales without implying causality or individual predictions.

Question 2

1. We are 95% confident that increasing ads by 1 will cause sales to increase by between \15 and \60.
2. There is a 95% probability that the true slope equals the midpoint of the interval.
3. We are 95% confident that for each additional online ad purchased, the mean daily sales in the population increase by between \15 and \60. (correct answer)
4. Because 0 is not in the interval, 95% of days will have sales between \15 and \60.
5. We are 95% confident that the correlation between ads and sales is between 15 and 60.

Explanation: This question tests interpreting a 95% confidence interval for the slope β in regressing sales on online ads. The interval (15, 60) indicates we are 95% confident that β, the average increase in mean daily sales per additional ad, lies between $15 and $60. Choice A distracts by implying causation from the interval, but confidence intervals do not confirm causal links. Choice E confuses the slope with correlation, which is unitless and between -1 and 1. Mini-lesson: A slope confidence interval encapsulates the uncertainty around the estimated population parameter β, representing mean change per unit; positive intervals suggest an increasing relationship, and the level like 95% means repeated sampling would capture β in 95% of such intervals, not that 95% of data points fall within it.

Question 3

1. Because 0 is in the interval, the correlation between sunlight and height is 0.
2. We are 98% confident that for each additional hour of sunlight, the population mean plant height changes by between -0.3 and 1.9 cm, on average. (correct answer)
3. There is a 98% chance that the true slope is between -0.3 and 1.9 for this sample.
4. About 98% of plants will grow between -0.3 and 1.9 cm for each extra hour of sunlight.
5. We are 98% confident that the correlation is between -0.3 and 1.9.

Explanation: This question examines interpretation when a confidence interval includes both positive and negative values. The interval (-0.3, 1.9) contains 0, meaning we cannot determine if the relationship is positive or negative at the 98% confidence level. Option B correctly interprets this: we are 98% confident that for each additional hour of sunlight, the population mean plant height changes by between -0.3 and 1.9 cm. Option A incorrectly concludes the correlation is exactly 0. Option C wrongly assigns probability to this sample's slope. Option D misapplies the interval to individual plants. Option E confuses slope with correlation values. Key insight: when an interval contains 0, the relationship could be positive, negative, or zero in the population.

Question 4

1. We are 95% confident that for each additional 1,000 emails sent, the mean weekly revenue in the population increases by between 0.0 and 2.5 thousand dollars. (correct answer)
2. Because the interval includes 0.0, the slope is exactly 0, so revenue and emails are uncorrelated.
3. There is a 95% probability that weekly revenue will increase by between \0 and \2{,}500 when 1,000 more emails are sent.
4. We are 95% confident that the correlation between emails and revenue is between 0.0 and 2.5.
5. Since the interval's lower endpoint is 0.0, it proves sending more emails cannot decrease revenue.

Explanation: This question examines interpreting a 95% confidence interval for the slope β of revenue on emails sent. The interval (0.0, 2.5) means we are 95% confident that β, the average increase in mean weekly revenue per 1,000 additional emails, is between $0 and $2,500. Choice B is a distractor, incorrectly asserting that including zero means the slope is exactly zero and variables are uncorrelated, but zero is just one plausible value. Choice C misapplies probability to individual revenue changes rather than the mean. Mini-lesson: Slope confidence intervals provide a range of feasible values for the population's average effect, with the lower bound at zero indicating non-negative plausibility; they do not prove directions or apply to correlations, and the confidence level pertains to the method's reliability over many samples, not single instances.

Question 5

1. We are 90% confident that for each 1$^\circ$F increase in temperature, the population mean ozone level decreases by between 0.4 and 1.8 ppb, on average. (correct answer)
2. There is a 90% probability that the true slope is negative.
3. Because the interval is negative, the correlation must be between -1.8 and -0.4.
4. About 90% of individual days will have ozone levels that drop by between 0.4 and 1.8 ppb for each 1$^\circ$F increase.
5. If we repeated the study many times, 90% of the time the sample slope would equal a value between -1.8 and -0.4 exactly.

Explanation: This question involves interpreting a confidence interval for slope when predicting ozone from temperature. The interval (-1.8, -0.4) is entirely negative, indicating an inverse relationship. Option A correctly states that we are 90% confident the population mean ozone level decreases by between 0.4 and 1.8 ppb for each 1°F increase in temperature. Option B incorrectly assigns probability to the parameter. Option C confuses slope values with correlation values (correlation must be between -1 and 1). Option D wrongly applies the interval to individual days rather than the population mean. Option E misunderstands what repeated sampling would show. Key insight: negative slopes indicate inverse relationships, and confidence intervals describe population parameters, not individual observations.

Question 6

1. Because 0 is in the interval, the slope is 0, so there is no relationship between sunlight and growth.
2. We are 99% confident that for each additional hour of sunlight, the mean weekly growth in the population changes by between $-0.4$ and $1.6$ cm. (correct answer)
3. There is a 99% chance that plants will grow between $-0.4$ and $1.6$ cm more each week for every extra hour of sunlight.
4. We are 99% confident that the correlation between sunlight and growth is between $-0.4$ and $1.6$.
5. Since the interval includes both negative and positive values, sunlight has no effect on growth for any plant.

Explanation: This question tests understanding a 99% confidence interval for the slope β in regressing plant growth on sunlight hours. The interval (-0.4, 1.6) suggests we are 99% confident that β, the average change in mean weekly growth per extra hour of sunlight, ranges from -0.4 to 1.6 cm. A frequent distractor is choice A, which erroneously concludes that including zero means the slope is exactly zero and no relationship exists, but it only means zero is plausible. Choice E overgeneralizes the interval's inclusion of negatives and positives to claim no effect for any plant, ignoring variability. Mini-lesson: Confidence intervals for slopes estimate the plausible range for the population's average response change per unit predictor increase; wider intervals at higher confidence levels reflect greater certainty, and overlapping zero indicates the data is consistent with no linear association without proving it.

Question 7

1. We are 90% confident that the correlation between humidity and maximum temperature is between $-0.30$ and $0.05$.
2. Because 0 is in the interval, the slope in the population must be 0.
3. There is a 90% chance that the true slope is between $-0.30$ and $0.05$ after seeing the data.
4. We are 90% confident that for each 1% increase in humidity, the mean maximum temperature in the population changes by between $-0.30$ and $0.05$ degrees F, on average. (correct answer)
5. 90% of individual days will have maximum temperature changes between $-0.30$ and $0.05$ degrees F for each 1% increase in humidity.

Explanation: This question involves a confidence interval (-0.30, 0.05) that contains zero. Option D correctly interprets this as being 90% confident that for each 1% increase in humidity, the mean maximum temperature changes by between -0.30 and 0.05 degrees F. Option A confuses slope with correlation, Option B incorrectly concludes the slope must be 0, Option C misinterprets confidence as posterior probability, and Option E applies the interval to individual days rather than the population mean. Key insight: when 0 is in the confidence interval, we cannot determine the direction of the relationship at that confidence level - the true slope could be positive, negative, or zero.

Question 8

1. We are 95% confident that the mean rent in the population decreases by between $20 and $85 for each additional mile from the city center, on average. (correct answer)
2. There is a 95% chance that the true slope is between $-85$ and $-20$ because this interval was computed from the sample.
3. We are 95% confident that $r$ is between $-85$ and $-20$.
4. Because the interval does not include 0, the rent must decrease by exactly $85 per mile in the population.
5. 95% of individual city rents will decrease by between $20 and $85 when distance increases by 1 mile.

Explanation: This question tests interpretation of a negative confidence interval for slope in an economics context. The interval (-85, -20) means we're 95% confident the true slope is between -85 and -20 dollars per mile. Option A correctly interprets this as the mean rent decreasing by between $20 and $85 for each additional mile from city center (note the positive phrasing of a negative relationship). Option B incorrectly assigns probability after seeing the data, Option C confuses slope with correlation, Option D makes an unfounded claim about the exact value, and Option E misapplies the interval to individual cities. Remember: confidence intervals describe our uncertainty about population parameters, not variability in individual observations.

Question 9

1. If fiber intake increases by 1 gram, then 95% of individuals will reduce LDL by between 0.2 and 1.9 mg/dL.
2. We are 95% confident that for each additional gram of fiber, the mean LDL cholesterol in the population decreases by between 0.2 and 1.9 mg/dL, on average. (correct answer)
3. There is a 95% chance that the slope is between $-1.9$ and $-0.2$ mg/dL per gram.
4. We are 95% confident that the correlation between fiber and LDL is between $-1.9$ and $-0.2$.
5. Because the interval is negative, fiber intake causes LDL to decrease for every individual.

Explanation: This question presents a negative confidence interval (-1.9, -0.2) in a health context. Option B correctly states we're 95% confident that for each additional gram of fiber, the mean LDL cholesterol decreases by between 0.2 and 1.9 mg/dL in the population. Option A incorrectly applies this to individual people, Option C misinterprets confidence as probability, Option D confuses slope with correlation (correlation has no units), and Option E makes an unfounded causal claim about every individual. Important distinction: regression describes associations on average, not deterministic relationships for every individual, and confidence intervals quantify our uncertainty about population parameters.

Question 10

1. We are 90% confident that for each additional mile from downtown, the mean rent in the population decreases by between \40 and \220 per month. (correct answer)
2. There is a 90% probability that the slope for this fitted line is between $-220$ and $-40$.
3. Because the interval contains negative values, the correlation must be between $-220$ and $-40$.
4. Since 0 is not in the interval, exactly 90% of neighborhoods farther from downtown have lower rent.
5. We are 90% confident that each additional mile causes rent to drop by between \40 and \220 per month.

Explanation: This question evaluates the interpretation of a 90% confidence interval for the slope β in a regression of rent on distance from downtown. The interval (-220, -40) means we are 90% confident that the true β, the average change in mean monthly rent per additional mile, is between -220 and -40 dollars, or a decrease of 40 to 220 dollars. Choice E is a distractor as it incorrectly assumes the interval implies causation, but confidence intervals do not establish cause-and-effect relationships. Choice C mistakenly equates the slope interval with the correlation coefficient, which is bounded between -1 and 1. Mini-lesson: A confidence interval for the regression slope provides a range where the true population average rate of change is likely to fall, accounting for sampling error; negative endpoints here indicate a plausible negative association, and the confidence level reflects the long-run success rate of the interval method in capturing β.

Question 11

1. Because 0 is in the interval, stretching has no association with 5K time in the population.
2. There is a 90% chance that the true slope is between $-0.25$ and $0.05$ minutes for each minute of stretching.
3. We are 90% confident that for each additional minute of stretching, the mean 5K time in the population changes by between $-0.25$ and $0.05$ minutes. (correct answer)
4. We are 90% confident that the correlation between stretching and 5K time is between $-0.25$ and $0.05$.
5. Since the interval includes negative values, 90% of runners will run faster by 0.25 minutes for each minute of stretching.

Explanation: This question tests the interpretation of a 90% confidence interval for the slope β relating stretching time to 5K completion time. The interval (-0.25, 0.05) suggests we are 90% confident that β, the average change in mean 5K time per additional minute of stretching, ranges from -0.25 to 0.05 minutes. A common distractor is choice A, which concludes no association because zero is included, but it only means no association is plausible, not proven. Choice E misinterprets negatives as guaranteeing faster times for most runners. Mini-lesson: Confidence intervals for regression slopes capture uncertainty in the average response change; intervals crossing zero are consistent with no linear effect, but they do not disprove associations or apply to correlations directly, emphasizing the need to distinguish population means from individual variations.

Question 12

1. There is a 95% chance that students who miss one more day will lose between 0.6 and 3.2 points on the exam.
2. We are 95% confident that for each additional absence, the population mean final exam score decreases by between 0.6 and 3.2 points, on average. (correct answer)
3. Because the interval is negative, the correlation between absences and exam score is between -3.2 and -0.6.
4. We are 95% confident that 95% of sample slopes will be negative.
5. There is a 95% probability that the true slope is -3.2 or -0.6.

Explanation: This question tests interpretation of a negative confidence interval in an educational setting. The interval (-3.2, -0.6) indicates that more absences are associated with lower exam scores. Option B correctly states that we are 95% confident that for each additional absence, the population mean final exam score decreases by between 0.6 and 3.2 points. Option A incorrectly applies this to individual students with certainty. Option C confuses slope values with correlation (correlation must be between -1 and 1). Option D makes an incorrect claim about sample slopes. Option E wrongly suggests the slope equals only the endpoints. Remember: confidence intervals describe our uncertainty about the true population relationship, not guarantees about individuals.

Question 13

1. There is a 99% chance that increasing caffeine by 1 mg will increase an individual person's heart rate by between 0.003 and 0.021 bpm.
2. We are 99% confident that for each additional 1 mg of caffeine intake, the mean resting heart rate increases by between 0.003 and 0.021 bpm in the population. (correct answer)
3. We are 99% confident that the correlation between caffeine intake and heart rate is between 0.003 and 0.021.
4. If we repeated the study many times, 99% of adults would have slopes between 0.003 and 0.021.
5. Because 0 is not in the interval, 99% of sample slopes will be between 0.003 and 0.021.

Explanation: The skill here is correctly interpreting a 99% confidence interval for the slope in a regression of heart rate on caffeine intake. The interval from 0.003 to 0.021 bpm per mg suggests we are 99% confident that the true average increase in resting heart rate per mg of caffeine is within these bounds. Choice A is a distractor, as it applies the interval to individual predictions rather than population means. Mini-lesson: A confidence interval for β means that the method captures the true slope in 99% of repeated samples; it doesn't give probabilities for individuals or the computed interval itself. Avoid confusing slope with correlation, and always specify the direction and units. Since zero is excluded, there's evidence of a positive relationship, but causation isn't implied by the interval alone.

Question 14

1. We are 98% confident that for each additional mile from downtown, the mean monthly rent decreases by between $30 and $120 in the population. (correct answer)
2. There is a 98% chance that any apartment 1 mile farther from downtown will rent for $30 to $120 less.
3. Because 0 is not in the interval, the correlation must be between -120 and -30.
4. If we repeated the sampling many times, 98% of the sample slopes would be negative, so the interval must be correct.
5. We are 98% confident that 98% of apartments have rents that decrease by $30 to $120 per mile from downtown.

Explanation: The skill involves interpreting a 98% confidence interval for the slope of rent on distance from downtown. The interval $(-120, -30)$ dollars per mile shows we are 98% confident that the true average decrease in rent per mile is between 30 and 120 dollars. Distractor B errs by applying the probability to individual apartments rather than the population mean. Mini-lesson: Slope intervals capture the plausible values for eta with the given confidence; higher levels widen the interval for more certainty. Always reference mean changes, not individuals, and avoid mixing with correlation. The negative range suggests a location-based rent gradient.

Question 15

1. We are 90% confident that for each additional mile of weekly training, the mean 5K time decreases by between 0.05 and 0.40 minutes in the population. (correct answer)
2. There is a 90% probability that the slope is negative, so 90% of runners will get faster when they add a mile of training.
3. We are 90% confident that the correlation between training mileage and 5K time is between 0.40 and 0.05.
4. Because the interval does not include 0, the slope equals 0.225 minutes per mile.
5. If the study were repeated many times, 90% of the time the true slope would change to fall between 0.40 and 0.05.

Explanation: This question assesses interpretation of a 90% confidence interval for the slope of 5K time on training mileage. The interval -0.40 to -0.05 minutes per mile means we are 90% confident that the true average decrease in time per additional mile trained is between 0.05 and 0.40 minutes. Choice C is a distractor, wrongly equating the slope interval with one for correlation. Mini-lesson: Confidence intervals for β reflect sampling error; 90% of them from repeats would include the true slope. Interpret as mean population effects with units, noting direction (here, negative for improvement). Excluding zero provides evidence against no association.

Question 16

1. We are 92% confident that the correlation between experience and salary is between 1500 and 4200.
2. There is a 92% chance that the true slope is between $1500$ and $4200$ because the confidence level is 92%.
3. We are 92% confident that for each additional year of experience, the mean annual salary increases by between $1500$ and $4200$ in the population. (correct answer)
4. Because 0 is not in the interval, 92% of employees will earn $1500$ to $4200$ more each year they work.
5. If many samples of 45 employees were taken, 92% of the time the sample slope would equal a value between $1500$ and $4200$ exactly.

Explanation: This question evaluates interpreting a 92% confidence interval for the slope of salary on work experience. The interval 1500 to 4200 dollars per year suggests we are 92% confident that the true average increase in salary per year of experience is within this range. A common distractor is choice A, which confuses the slope (with units) with the unitless correlation. Mini-lesson: Confidence intervals for slopes provide a range for β, where the level indicates long-run success rate of capturing the true value. Always specify mean population effects, units, and direction; excluding zero supports a positive association. Avoid applying to individuals or claiming probabilities for the fixed interval.

Question 17

1. We are 90% confident that for each 10F increase in daily high temperature, the mean electricity use changes by between 1.5 and 0.4 kWh in the population. (correct answer)
2. There is a 90% probability that the true slope is negative because most of the interval is below 0.
3. Because 0 is in the interval, the slope must be 0, so temperature and electricity use are unrelated.
4. We are 90% confident that the correlation between temperature and electricity use is between 1.5 and 0.4.
5. In 90% of all samples of 30 days, the true slope will fall between 1.5 and 0.4.

Explanation: This question assesses understanding of confidence intervals for the regression slope relating temperature to electricity use. The 90% confidence interval spans -1.5 to 0.4 kWh per °F, indicating we are 90% confident that the true population slope, or average change in electricity use per degree increase, lies in this range, which includes both negative and positive values. A frequent distractor is choice C, which wrongly concludes that including zero means the slope is exactly zero and variables are unrelated. Mini-lesson: Confidence intervals for slopes account for sampling variability and provide a plausible range for β; if zero is included, we lack evidence against no relationship, but it doesn't prove the slope is zero. Interpretations should reference the mean population effect, not probabilities for the fixed interval or correlations. Note the units to clarify the practical meaning.

Question 18

1. There is a 90% chance that the sample slope will fall between 0.002 and 0.006 if the same people are measured again.
2. We are 90% confident that for each additional 1 mg of sodium intake, the mean systolic blood pressure increases by between 0.002 and 0.006 mmHg in the population of adults like those sampled. (correct answer)
3. Since the interval is positive, 90% of individuals' blood pressures increase by 0.002 to 0.006 mmHg for each 1 mg increase in sodium.
4. We are 90% confident that the correlation between sodium intake and systolic blood pressure is between 0.002 and 0.006.
5. Because 0 is not included, sodium intake is proven to cause higher blood pressure.

Explanation: This question tests understanding of confidence intervals for slope in a health context. The interval (0.002, 0.006) represents the change in blood pressure per mg of sodium. Choice B correctly interprets this as being 90% confident about the mean increase in systolic blood pressure per mg of sodium for the population. Choice A incorrectly refers to repeated sampling of the same people. Choice C misapplies the interval to individual responses. Choice D confuses slope with correlation. Choice E incorrectly claims causation. A confidence interval for slope estimates the average linear relationship in the population, not causal effects or individual responses.

Question 19

1. We are 90% confident that each additional mile from downtown is associated with a decrease of between $20 and $85 in the mean monthly rent, on average. (correct answer)
2. There is a 90% probability that the slope is negative because the interval is below 0.
3. Because the interval is negative, 90% of apartments will have rents that drop by between $20 and $85 for each mile farther from downtown.
4. We are 90% confident that the correlation between distance and rent is between $-85$ and $-20$.
5. If many samples were taken, 90% of the sample slopes would equal the true slope and fall between $-85$ and $-20$.

Explanation: This question involves interpreting a confidence interval for slope when the relationship is negative. The interval (-85, -20) indicates we're 90% confident the true slope lies in this range. Choice A correctly states that each additional mile from downtown is associated with a decrease (negative slope) of between $20 and $85 in mean monthly rent. Choice B incorrectly treats the confidence level as a probability about the slope being negative. Choice C wrongly applies the interval to individual apartments rather than the mean. Choice D confuses slope with correlation values. Choice E misunderstands how confidence intervals work across repeated sampling. Key insight: negative slopes indicate inverse relationships, and we interpret the magnitude of change.

Question 20

1. Because 0 is in the interval, there is definitely no association between practice sessions and free-throw percentage.
2. We are 90% confident that for each additional practice session, the mean free-throw percentage changes by between $-0.5$ and $2.0$ percentage points, on average. (correct answer)
3. There is a 90% chance that the correlation between sessions and free-throw percentage is between $-0.5$ and $2.0$.
4. We are 90% confident that 90% of players will improve by between $-0.5$ and $2.0$ percentage points for each extra session.
5. If we repeated the study many times, 90% of the time the true slope would vary and fall in this interval.

Explanation: This question involves interpreting a confidence interval that contains zero. The interval (-0.5, 2.0) includes 0, meaning we cannot conclude there's a significant relationship at the 10% level. Choice B correctly interprets this: we're 90% confident the true slope (change in mean free-throw percentage per session) lies between -0.5 and 2.0 percentage points. Choice A overstates the conclusion - we can't definitively say there's no association. Choice C confuses slope with correlation. Choice D wrongly applies this to individual players with a specific probability. Choice E misunderstands how confidence intervals relate to the true parameter. Key point: intervals containing zero suggest the relationship might be positive, negative, or nonexistent.