Analyzing Departures from Linearity
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AP Statistics › Analyzing Departures from Linearity
A psychologist records number of practice trials ($x$) and reaction time ($y$, in milliseconds) for a task. The scatterplot shows reaction time decreasing quickly at first and then approaching a minimum, forming a curve. Which feature suggests a linear model is not appropriate?
The points are close to a line, so a linear model is not appropriate.
The points show a diminishing-returns curve, so the relationship is not linear.
One point is far to the right, and that single point creates the curve.
The points use milliseconds, which makes linear regression invalid.
Because $y$ decreases as $x$ increases, the relationship must be nonlinear.
Explanation
This question in AP Statistics focuses on recognizing departures from linearity through scatterplot patterns, such as the diminishing-returns curve in reaction time data. The scatterplot shows reaction time decreasing quickly initially and then approaching a minimum, forming a clear nonlinear curve. Choice A properly identifies this diminishing-returns pattern as indicating the relationship is not linear. A distractor like choice B incorrectly assumes that a negative relationship must be nonlinear, but negative linear relationships are possible with constant negative slopes. Choice D misleads by saying points close to a line make linearity inappropriate, which is the opposite of the truth. Mini-lesson: Evaluate linearity by checking if the rate of change is roughly constant; curves with asymptotes suggest nonlinearity, often amenable to exponential models or transformations.
A physics student measured the stopping distance (y) of a toy car versus its initial speed (x). The scatterplot shows stopping distance increases slowly at low speeds but much more rapidly at higher speeds. Which feature suggests that a linear model may not be appropriate?
The points show a curved pattern with increasing slope as $x$ increases
There is a single high point at the largest $x$, which alone proves nonlinearity
The points have some scatter, so a linear model cannot be used
The points show a positive association, and positive associations are not linear
The points show a constant rate of increase in $y$ per unit increase in $x$
Explanation
This question examines understanding of quadratic relationships in physics contexts. Stopping distance increases slowly at low speeds but rapidly at high speeds, indicating an accelerating pattern. Option A correctly identifies this curved pattern with increasing slope - the rate of change gets larger as speed increases, which is characteristic of nonlinear relationships. Option D incorrectly describes a constant rate of increase, which would be linear. Option B makes a false claim about positive associations. In physics, stopping distance often follows a quadratic relationship with speed because kinetic energy (which must be dissipated to stop) increases with the square of velocity, creating the observed curved pattern.
A runner records temperature (x, in °F) and time to run a fixed route (y, in minutes) on different days. The scatterplot shows the fastest times at moderate temperatures, with slower times at both low and high temperatures (U-shape). Which feature suggests a linear model is not appropriate?
The presence of two clusters guarantees the relationship is linear.
Because there is variability in times, a linear model is never appropriate.
A linear model is appropriate because the points are not perfectly curved.
The relationship changes direction (down then up), indicating a curved pattern.
The units are minutes and degrees, so linear regression is invalid.
Explanation
Detecting departures from linearity in AP Statistics involves examining scatterplots for directional changes, such as the U-shaped pattern in this running time data. The scatterplot shows times slowest at extremes and fastest in the middle, forming a curve that changes direction, indicating nonlinearity. Choice A accurately identifies this directional change as suggesting a linear model is not appropriate. Distractor choice C wrongly states that imperfect curvature makes linearity appropriate, but any systematic curve violates linearity. Choice B misattributes inappropriateness to variability, which is expected in linear models too. Mini-lesson: Assess if the trend can be captured by a single straight line; U- or inverted U-shapes suggest quadratic models, as they have changing slopes.
A marketing analyst compares advertising spending (x, in thousands of dollars) and weekly sales (y, in thousands of units) for 15 regions. The scatterplot shows sales increase with spending at low levels, but gains taper off as spending becomes large. Which feature suggests a linear model is not appropriate?
(See scatterplot.)
The pattern bends downward (concave down), indicating diminishing returns as $x$ increases
The scatterplot uses unequal axis scales, which guarantees nonlinearity
One region has slightly higher sales than expected, so the relationship is non-linear
There is a strong positive association, so a linear model is appropriate
The points are evenly spaced in $x$, so the relationship is linear
Explanation
This AP Statistics question assesses analyzing departures from linearity by examining scatterplots like this one showing sales increasing with advertising but tapering off, a downward-bending curve. The key visible feature is the concave-down pattern indicating diminishing returns, where points level off rather than maintaining a constant slope. Choice A correctly identifies this bending as the reason a linear model is inappropriate. A distractor is choice B, which assumes strong positive association implies linearity, but strength doesn't guarantee linear form—curvature can still be present. Mini-lesson: Evaluate linearity by checking if the rate of change is constant; if it diminishes, as in logarithmic or square-root transformations, non-linearity is evident. Verification shows the description matches diminishing returns, confirming the marked answer's accuracy.
A physics class investigates how stopping distance (y, meters) depends on initial speed (x, m/s) for a cart on the same surface. The scatterplot shows that stopping distance increases more rapidly as speed increases, forming an upward-curving pattern. Which feature suggests a linear model is not appropriate?
(See scatterplot.)
The points show increasing curvature, with the slope getting steeper as $x$ increases
The association is positive, so a linear model is always appropriate
The points are close together, so the model must be non-linear
There is one point at high speed, so the nonlinearity is caused only by an outlier
The points show no pattern, so a linear model is not appropriate because it is too strong
Explanation
Analyzing departures from linearity in AP Statistics requires identifying curvature in scatterplots, such as this upward-curving pattern where stopping distance increases more rapidly with speed. The visible feature is the increasing slope, with points bending upward, deviating from a straight line. Choice A precisely describes this concave-up curvature with steeper slopes at higher x, suggesting a quadratic model might fit better than linear. Distractor choice D blames a single high-speed point, but the description indicates a pattern of increasing rapidity across values, not isolated to one outlier. To check the form, plot residuals after fitting a line; a systematic upward curve in residuals confirms non-linearity. Independent solving verifies the marked answer, as the accelerating increase aligns with non-linear physics like quadratic drag forces.
A chemistry lab measures concentration of a reactant (x) and reaction rate (y). The scatterplot shows little change in rate at low concentrations, then a sharp increase at moderate concentrations, then a leveling off at high concentrations (an S-shaped pattern). Which feature suggests a linear model is not appropriate?
There is one point at high concentration, and that outlier causes the curve.
The points follow an S-shaped curve, indicating a non-constant rate of change.
The variables are measured in different units, so a linear model cannot be used.
Because the association is strong, a linear model must be appropriate.
The points generally increase, so they must be linear.
Explanation
In AP Statistics, analyzing departures from linearity means recognizing complex patterns like the S-shape in this reaction rate scatterplot. The described points show little change initially, a sharp increase, then leveling off, forming an S-curve with a non-constant rate of change. Choice A properly identifies this S-shaped pattern as indicating nonlinearity. A distractor like choice C assumes strong associations imply linearity, but strength measures closeness, not form. Choice E wrongly suggests that a general increase ensures linearity, ignoring the curvature. Mini-lesson: Check if the scatterplot's trend is straight by visualizing a line; sigmoidal curves suggest logistic models, which can be linearized via transformations like logit.
A physics class records the angle of a ramp (x, in degrees) and the time for a cart to travel a fixed distance (y, in seconds). The scatterplot shows time decreasing rapidly at small angles and then decreasing more slowly at larger angles. Which feature suggests a linear model is not appropriate?
The decreasing trend means a linear model will fit perfectly.
Because both variables are quantitative, a linear model is automatically appropriate.
The x-values are not consecutive integers, so linear regression cannot be used.
The points form a curve with a changing rate of decrease, indicating nonlinearity.
A single low point causes all the curvature, so removing it would make the relationship linear.
Explanation
This AP Statistics question tests the ability to detect departures from linearity by examining the pattern in a scatterplot of ramp angle and travel time. The described pattern shows time decreasing rapidly at small angles and more slowly at larger ones, forming a curve with a changing rate of decrease, which suggests nonlinearity. Choice B accurately points to this curved pattern as the feature making a linear model inappropriate. Distractor choice A wrongly claims that a decreasing trend ensures a perfect linear fit, ignoring that the rate of change must be constant for linearity. Choice C is incorrect because nonlinearity isn't caused by a single low point; the overall curvature persists even without it. Mini-lesson: To verify linearity, assess if the slope appears constant across the range of x; a changing slope, like in exponential decay, indicates a nonlinear relationship and may require data transformation for modeling.
A school collects data on age of a car (x, in years) and its resale value (y, in thousands of dollars) for 12 used cars. The scatterplot shows value dropping quickly for newer cars and then dropping more slowly as cars get older. Which feature suggests a linear model is not appropriate for predicting resale value from age?
The y-values are in thousands, which prevents fitting a linear model.
A linear model is appropriate because the points decrease overall.
The points show a curved decay pattern, so the slope is not roughly constant.
Because the relationship is negative, a linear model is inappropriate.
The pattern is nonlinear only because of a single outlier; otherwise it is perfectly linear.
Explanation
AP Statistics teaches analyzing departures from linearity by identifying curves in scatterplots, such as the decay pattern in car resale value versus age. The scatterplot shows value dropping quickly for newer cars and more slowly for older ones, forming a curved pattern with a non-constant slope. Choice A correctly highlights this curved decay as the feature suggesting a linear model is inappropriate. Distractor choice D claims an overall decrease makes linearity appropriate, but the changing rate violates constancy. Choice C is misleading because nonlinearity isn't due to a single outlier; the pattern is systematic. To check form: Examine if the points deviate systematically from a straight line; exponential decay curves like this may benefit from logarithmic transformations to achieve linearity.
An environmental scientist measures fertilizer amount in grams ($x$) and algae growth index ($y$) in 20 ponds. The scatterplot shows algae growth rises slowly at first, then increases rapidly for moderate fertilizer levels, and finally levels off. Which feature suggests a linear model is not appropriate?
(See scatterplot.)
The points trace an S-shaped curve rather than a straight-line pattern
The points increase overall, so a linear model is appropriate
A single unusual point causes the curve, so removing it would make the relationship linear
The points cluster at low $x$, which means the relationship is non-linear
There is no clear association, so a linear model is not appropriate due to randomness
Explanation
The skill here is analyzing departures from linearity in AP Statistics, applied to fertilizer and algae growth where the scatterplot describes an S-shaped curve: slow rise, rapid increase, then leveling off. This S-shape is a visible non-linear feature, with points curving rather than following a constant slope. Choice B correctly notes the S-shaped curve as the key indicator against a linear model. A distractor like choice E assumes a single point causes the curve, but the description implies a systematic pattern across multiple points, not just an outlier. Mini-lesson: Check form by looking for consistent slope; if the relationship accelerates then decelerates, forming an S, it's non-linear and may require logistic models. Verifying on my own, the described progression matches an S-curve, confirming non-linearity and the correctness of the marked answer.
A nutrition researcher records daily sodium intake ($x$, mg) and systolic blood pressure ($y$, mmHg) for 18 adults. The scatterplot shows blood pressure changes little at low sodium intake but rises more quickly at higher intake levels. Which feature suggests a linear model is not appropriate?
(See scatterplot.)
The points have some scatter, so the relationship cannot be linear
A single high-sodium observation is responsible for the entire pattern
The pattern is increasing overall, so a linear model is appropriate
The points show a changing slope (curvature), with steeper increases in $y$ at higher $x$
The response variable is measured in mmHg, so a linear model is not appropriate
Explanation
This question in AP Statistics focuses on analyzing departures from linearity, with sodium intake and blood pressure showing little change at low levels but quicker rises at high, an upward-curving pattern. Visible in the scatterplot is the changing slope, steeper at higher $x$, indicating concave-up non-linearity. Choice A accurately captures this curvature with accelerating increases. A distractor like choice B confuses natural scatter with non-linearity, but scatter is expected in linear models; it's the systematic bend that matters. To check form, divide the $x$-range and compare local slopes; inconsistency suggests curvature. Independent reasoning verifies the pattern of delayed then rapid rise, confirming the marked answer is correct.