Confidence Intervals: Slope of Regression Models - AP Statistics

A psychologist fits a regression line predicting stress score from hours of sleep per night for a random sample of adults. A 95% confidence interval for the population slope is $(-2.8,\ -0.4)$ . Which interpretation is correct?
Answer: We are 95% confident that for each additional hour of sleep, the mean stress score decreases by between 0.4 and 2.8 points in the population of adults like those sampled. This question tests understanding of a negative confidence interval for slope in a psychological context. The interval (-2.8, -0.4) indicates stress scores decrease with more sleep. Choice B correctly interprets this as being 95% confident that mean stress score decreases by 0.4 to 2.8 points per hour of sleep for the population. Choice A incorrectly applies this to each individual person. Choice C confuses slope with correlation. Choice D incorrectly uses probability language about the slope being negative. Choice E incorrectly claims the data prove causation. A confidence interval for slope estimates the population parameter describing the average relationship, not individual effects or causal relationships.
A company analyzes whether the number of service calls (per month) predicts customer satisfaction score (0–100) using a random sample of 60 customers. A 95% confidence interval for the slope is $(-0.50,\ 0.20)$ satisfaction points per call. Which interpretation is correct?
Answer: We are 95% confident that for each additional service call, the population mean satisfaction changes by between -0.50 and 0.20 points, on average. This question examines interpretation when a confidence interval contains zero. The interval (-0.50, 0.20) for the slope includes 0, which means we cannot conclude whether the relationship is positive or negative. Option C correctly interprets this: we are 95% confident that for each additional service call, the population mean satisfaction changes by between -0.50 and 0.20 points. Option A confuses slope with correlation. Option B incorrectly concludes no relationship exists (we simply can't determine direction). Option D wrongly assigns probability to the parameter equaling exactly 0. Option E misapplies the interval to individual predictions. Important concept: when a confidence interval for slope contains 0, we cannot reject the null hypothesis of no linear relationship at the corresponding significance level.
A school district collected data from a random sample of 32 high schools relating average weekly homework time (hours, $x$ ) to average SAT Math score ( $y$ ). A least-squares regression line was fit to predict $y$ from $x$ . A 95% confidence interval for the true slope of the regression line is $(4.2,\ 11.6)$ SAT points per hour. Which interpretation is correct?
Answer: We are 95% confident that for each additional hour of homework per week, the mean SAT Math score for the population of similar high schools increases by between 4.2 and 11.6 points, on average. This question tests understanding of confidence intervals for the slope of a regression line. The interval (4.2, 11.6) represents our confidence about the true population slope, not the sample slope we already calculated. Option B correctly interprets this as being 95% confident that for each additional hour of homework, the mean SAT Math score increases by between 4.2 and 11.6 points in the population. Option A incorrectly refers to the sample slope (which is a fixed value, not probabilistic), Option C misinterprets this as applying to individual students rather than the mean, Option D confuses slope with correlation, and Option E misunderstands confidence intervals entirely. Remember: confidence intervals for slopes tell us about the average change in y for a one-unit change in x across the entire population.
A biologist measures 20 plants of the same species, recording $x$ = hours of sunlight per day and $y$ = weekly growth (cm). The regression of growth on sunlight yields a 99% confidence interval for the population slope $\beta$ of $(-0.4,\ 1.6)$ . Which interpretation is correct?
Answer: 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. 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.
A manufacturing engineer samples 30 production runs and records machine calibration setting ( $x$ ) and average defect rate ( $y$ , defects per 1,000 units). A least-squares regression line predicts defect rate from calibration setting. A 95% confidence interval for the true slope is $(1.2,\ 3.5)$ defects per 1,000 units per calibration unit. Which interpretation is correct?
Answer: We are 95% confident that for each 1-unit increase in calibration setting, the mean defect rate in the population increases by between 1.2 and 3.5 defects per 1,000 units, on average. This question presents a positive confidence interval (1.2, 3.5) in a manufacturing context. Option C correctly states we're 95% confident that for each 1-unit increase in calibration setting, the mean defect rate increases by between 1.2 and 3.5 defects per 1,000 units in the population. Option A incorrectly applies this to every individual run, Option B misinterprets confidence as probability, Option D confuses slope with correlation, and Option E makes an unjustified causal claim. Important: confidence intervals describe statistical associations, not causal relationships - correlation does not imply causation even when the interval doesn't contain zero.
A real estate analyst fits a regression line with home sale price (in thousands of dollars) as the response and home size (in square feet) as the explanatory variable, using a random sample of recent sales in a city. A 95% confidence interval for the population slope is $(0.08,\ 0.14)$ . Which interpretation is correct?
Answer: We are 95% confident that for each additional square foot, the mean sale price increases by between $0.08$ and $0.14$ thousand dollars for homes in the population similar to those sampled. This question involves interpreting a confidence interval for slope in a real estate context. The interval (0.08, 0.14) represents thousands of dollars per square foot. Choice A correctly interprets this as being 95% confident that mean sale price increases by $80 to $140 per square foot in the population. Choice B incorrectly claims the true slope is exactly 0.11. Choice C misapplies the interval to individual homes rather than the population mean. Choice D confuses slope with correlation. Choice E makes an incorrect guarantee about individual homes. The slope represents the average relationship in the population, and the confidence interval quantifies our uncertainty about this parameter.
A teacher investigates whether the number of absences predicts final exam score (out of 100) using a random sample of 30 students. A 95% confidence interval for the slope is $(-3.2,\ -0.6)$ points per absence. Which interpretation is correct?
Answer: 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. 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.
An environmental scientist models ozone level (ppb) as a function of daily high temperature ( $^\circ$ F) using data from 25 randomly selected days. A 90% confidence interval for the regression slope is $(-1.8,\ -0.4)$ ppb per $^\circ$ F. Which interpretation is correct?
Answer: 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. 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.
An environmental scientist models the relationship between daily high temperature ( $x$ , in $^0$ F) and electricity use ( $y$ , in kWh) for 30 randomly selected days. A 90% confidence interval for the population slope is $(-1.5, 0.4)$ kWh per $^0$ F. Which interpretation is correct?
Answer: 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. 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.
A city planner studies whether distance from downtown ( $x$ miles) predicts monthly rent ( $y$ dollars) using 48 randomly selected apartments. A 98% confidence interval for the population slope is $(-120, -30)$ dollars per mile. Which interpretation is correct?
Answer: We are 98% confident that for each additional mile from downtown, the mean monthly rent decreases by between $30 and $120 in the population. 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 $\beta$ 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.
An environmental scientist models ozone level (ppb) as a linear function of daily temperature (°F) using data from randomly selected summer days in a city. A 90% confidence interval for the slope of the regression line is $(-0.7,\ 1.9)$ . Which interpretation is correct?
Answer: We are 90% confident that for each 1°F increase in temperature, the mean ozone level changes by between −0.7 and 1.9 ppb in the population of days like those sampled. This question involves interpreting a confidence interval for slope that contains zero. The interval (-0.7, 1.9) means we're 90% confident the true population slope lies in this range. Choice A correctly states this as the mean change in ozone level per degree Fahrenheit for the population. Choice B incorrectly refers to the sample slope rather than the population parameter. Choice C makes an incorrect definitive claim about no relationship - containing zero means we can't conclude a significant relationship exists, not that none exists. Choice D confuses slope with correlation. Choice E misinterprets the interval as referring to individual ozone levels. When a confidence interval contains zero, we cannot conclude there's a significant linear relationship at that confidence level.
A school counselor collects data from 28 students on $x$ = number of hours of sleep on a school night and $y$ = score on a memory quiz (out of 50). A least-squares regression of quiz score on sleep produces a 95% confidence interval for the population slope $\beta$ of $(1.2,\ 3.8)$ . Which interpretation is correct?
Answer: We are 95% confident that for each additional hour of sleep, the mean memory quiz score in the population increases by between 1.2 and 3.8 points. This question assesses the skill of interpreting a 95% confidence interval for the slope β in a linear regression model relating hours of sleep to memory quiz scores. The interval (1.2, 3.8) indicates that we are 95% confident the true population slope β, representing the average increase in mean quiz score per additional hour of sleep, lies between 1.2 and 3.8 points. A common distractor is choice A, which incorrectly treats the confidence interval as a probability statement about β for the specific sample rather than the population parameter. Another distractor, choice E, wrongly infers causality from the positive interval, but regression does not prove causation. In a mini-lesson on slope confidence intervals: these intervals capture the range of plausible values for β with a specified confidence level, based on sample data and variability; they apply to the mean change in the response variable, not individual outcomes, and excluding zero suggests evidence of a linear relationship.
A fitness researcher randomly selects 40 adults and measures resting heart rate ( $y$ , beats per minute) and weekly minutes of moderate exercise ( $x$ ). A least-squares regression line predicts $y$ from $x$ . A 99% confidence interval for the true slope is $(-0.090,\ -0.015)$ bpm per minute of exercise. Which interpretation is correct?
Answer: We are 99% confident that for each additional minute of weekly exercise, the mean resting heart rate in the population decreases by between 0.015 and 0.090 bpm, on average. This question presents a confidence interval that is entirely negative: (-0.090, -0.015). The correct interpretation (Option B) states we're 99% confident that for each additional minute of exercise, the mean resting heart rate decreases by between 0.015 and 0.090 bpm in the population. Option A incorrectly applies this to individual people rather than population means, Option C confuses slope with correlation (correlation is unitless and bounded between -1 and 1), Option D makes an incorrect claim about sample slopes, and Option E misunderstands what confidence intervals contain. Important: a negative slope means as x increases, y decreases on average, and confidence intervals always refer to population parameters, not individual predictions.
A school district examines whether class size ( $x$ , students) predicts average math test score ( $y$ , points) using a random sample of 25 classes. A 95% confidence interval for the population slope is $(-1.8, -0.2)$ points per student. Which interpretation is correct?
Answer: We are 95% confident that as class size increases by 1 student, the mean math score decreases by between 0.2 and 1.8 points in the population. This question evaluates interpretation of a confidence interval for the slope linking class size to math scores. The 95% interval from -1.8 to -0.2 points per student indicates we are 95% confident that the true population slope, or average decrease in math score per additional student, is between these negative values. Distractor D confuses the slope interval with one for correlation, which is unitless and bounded between -1 and 1. Mini-lesson: Slope confidence intervals estimate β, the mean change in Y per unit X, with the confidence level reflecting long-run coverage of the true value. Excluding zero suggests a significant negative association, but don't extend this to individual outcomes. Use the endpoints to describe the range of plausible effects in context.
A real estate analyst studies the relationship between house size ( $x$ , in square feet) and selling price ( $y$ , in thousands of dollars) for 35 homes. A 98% confidence interval for the true slope is $(0.04,\ 0.11)$ thousand dollars per square foot. Which interpretation is correct?
Answer: We are 98% confident that for each additional square foot, the mean selling price increases by between $0.04$ and $0.11$ thousand dollars, on average. This question involves interpreting a positive slope confidence interval in a real estate context. The interval (0.04, 0.11) means we're 98% confident the true slope lies in this range. Choice A correctly interprets this: for each additional square foot, the mean selling price increases by between $40 and $110 (since units are thousands of dollars). Choice B wrongly applies this to individual homes with certainty. Choice C confuses slope with correlation values. Choice D misunderstands confidence intervals by suggesting the parameter changes. Choice E incorrectly describes the sampling distribution of slopes. Key insight: always pay attention to units when interpreting slopes - here it's thousands of dollars per square foot.
A farmer models the relationship between fertilizer amount ( $x$ , in kilograms) and crop yield ( $y$ , in bushels) for 32 fields. A 92% confidence interval for the true slope is $(0.8,\ 1.6)$ bushels per kilogram. Which interpretation is correct?
Answer: We are 92% confident that for each additional kilogram of fertilizer, the mean crop yield increases by between 0.8 and 1.6 bushels, on average. This question involves interpreting a positive slope confidence interval in an agricultural context. The interval (0.8, 1.6) tells us we're 92% confident the true slope parameter lies between these values. Choice B correctly states that for each additional kilogram of fertilizer, the mean crop yield increases by between 0.8 and 1.6 bushels. Choice A incorrectly treats the confidence level as a probability about the parameter. Choice C wrongly applies this to a percentage of fields with certainty. Choice D confuses slope with correlation (correlation is unitless and bounded). Choice E misunderstands confidence intervals by suggesting the true parameter varies. Remember: confidence intervals quantify our uncertainty about fixed but unknown population parameters.
A botanist uses least-squares regression to predict plant height ( $y$ , cm) from amount of fertilizer ( $x$ , grams) using a random sample of 20 plants grown under similar conditions. A 95% confidence interval for the population slope is $( -0.6, 1.4)$ cm per gram. Which interpretation is correct?
Answer: We are 95% confident that for each additional gram of fertilizer, the mean plant height changes by between 0.6 and 1.4 cm in the population. The task is to interpret a 95% confidence interval for the slope of plant height on fertilizer amount. The interval -0.6 to 1.4 cm per gram indicates we are 95% confident that the true average change in height per gram of fertilizer could be negative or positive within this range, including zero. Distractor A incorrectly claims that including zero means no effect for any plant, ignoring variability. Mini-lesson: Such intervals estimate β; including zero means insufficient evidence to reject no relationship. Focus on mean changes, not individuals or probabilities for the interval itself, and distinguish from correlation. This setup highlights uncertainty in the relationship.
A nutrition scientist samples 22 adults and measures daily fiber intake ( $x$ , grams) and LDL cholesterol ( $y$ , mg/dL). A least-squares regression line predicts LDL from fiber intake. A 95% confidence interval for the true slope is $(-1.9,\ -0.2)$ mg/dL per gram. Which interpretation is correct?
Answer: 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. 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.
A sports scientist fits a least-squares regression line to predict 5K race time ( $y$ , minutes) from average weekly training mileage ( $x$ , miles) using data from 28 randomly selected runners. A 90% confidence interval for the population slope is $(-0.40, -0.05)$ minutes per mile. Which interpretation is correct?
Answer: 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. 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.
A biologist uses linear regression to predict plant height (cm) from hours of sunlight per day for a random sample of 22 plants. A 98% confidence interval for the slope is $(-0.3,\ 1.9)$ cm per hour. Which interpretation is correct?
Answer: 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. 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.

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