AP Statistics Quiz: Analyzing Departures From Linearity

What this quiz covers

This quiz focuses on Analyzing Departures From Linearity, 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. The points show a diminishing-returns curve, so the relationship is not linear. (correct answer)
2. Because $y$ decreases as $x$ increases, the relationship must be nonlinear.
3. One point is far to the right, and that single point creates the curve.
4. The points are close to a line, so a linear model is not appropriate.
5. The points use milliseconds, which makes linear regression invalid.

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.

Question 2

1. The points show a curved pattern with increasing slope as $x$ increases (correct answer)
2. The points show a positive association, and positive associations are not linear
3. There is a single high point at the largest $x$, which alone proves nonlinearity
4. The points show a constant rate of increase in $y$ per unit increase in $x$
5. The points have some scatter, so a linear model cannot be used

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.

Question 3

1. The points follow an S-shaped curve, indicating a non-constant rate of change. (correct answer)
2. There is one point at high concentration, and that outlier causes the curve.
3. Because the association is strong, a linear model must be appropriate.
4. The variables are measured in different units, so a linear model cannot be used.
5. 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.

Question 4

1. The decreasing trend means a linear model will fit perfectly.
2. The points form a curve with a changing rate of decrease, indicating nonlinearity. (correct answer)
3. A single low point causes all the curvature, so removing it would make the relationship linear.
4. Because both variables are quantitative, a linear model is automatically appropriate.
5. The x-values are not consecutive integers, so linear regression cannot be used.

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.

Question 5

1. The points show a curved decay pattern, so the slope is not roughly constant. (correct answer)
2. Because the relationship is negative, a linear model is inappropriate.
3. The pattern is nonlinear only because of a single outlier; otherwise it is perfectly linear.
4. A linear model is appropriate because the points decrease overall.
5. The y-values are in thousands, which prevents fitting a linear model.

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.

Question 6

1. The pattern rises then falls, showing curvature rather than a straight-line trend. (correct answer)
2. There is one extreme x-value, so the relationship must be nonlinear.
3. The association is moderate, so a linear model is always appropriate.
4. The points are scattered, which guarantees the relationship is linear.
5. The y-values are measured in centimeters, which prevents linear modeling.

Explanation: In AP Statistics, analyzing departures from linearity involves inspecting scatterplots for patterns that deviate from a straight line, such as the inverted U-shape described here. The scatterplot shows plant height increasing with fertilizer amount initially and then decreasing, forming a curved pattern that rises and falls, which clearly indicates nonlinearity. Choice A correctly identifies this curvature as the reason a linear model is inappropriate for prediction. A distractor like choice B incorrectly attributes nonlinearity solely to an extreme x-value, but the overall pattern, not just one point, determines the form. Choice C is misleading because a moderate association does not automatically make a linear model suitable if the pattern is curved. For a mini-lesson on checking form: overlay an imaginary straight line on the scatterplot; if the points systematically deviate (e.g., following a parabola), it's nonlinear, and transformations or polynomial models may be needed.

Question 7

1. There is a strong positive association, so a straight line will always fit well.
2. The points follow a curved pattern that levels off, indicating nonlinearity. (correct answer)
3. One point is slightly higher than the others, so the relationship is nonlinear.
4. The points are tightly clustered, which suggests a linear model is inappropriate.
5. The x-values are evenly spaced, which prevents using a linear model.

Explanation: This question assesses the skill of analyzing departures from linearity in scatterplots, a key concept in AP Statistics for determining appropriate models. The scatterplot described shows quiz scores rising quickly initially and then leveling off as study hours increase, which is a visible curved pattern indicating nonlinearity. The correct answer, choice B, identifies this leveling-off curvature as the feature suggesting a linear model is not appropriate. A common distractor, like choice A, mistakenly assumes that a strong positive association guarantees a good linear fit, but strength alone does not ensure linearity. Another distractor, choice C, focuses on a single point being higher, but nonlinearity is determined by the overall pattern, not isolated points. To check the form of a relationship, examine if the scatterplot follows a roughly straight line; systematic curvature, such as diminishing returns, indicates a departure from linearity and suggests exploring nonlinear models like quadratic or exponential transformations.

Question 8

1. The points show a curved pattern with a leveling-off effect rather than a straight-line trend. (correct answer)
2. There is no outlier, so a linear model must be appropriate.
3. The association is negative, and negative associations cannot be modeled linearly.
4. Because rent is measured in dollars, the relationship cannot be linear.
5. A linear model is appropriate because the points generally go down as x increases.

Explanation: Analyzing departures from linearity in AP Statistics requires identifying when scatterplots show systematic curves rather than straight trends, as in this rent versus distance example. The scatterplot depicts rent dropping steeply near downtown and then flattening, a visible leveling-off curve that deviates from linearity. Choice A correctly highlights this curved pattern with a leveling-off effect as the reason a linear model is not suitable. Distractor choice E suggests that a general downward trend guarantees linearity, but direction alone doesn't confirm a constant slope. Choice C is wrong because negative associations can be linear, but here the curvature violates that assumption. For checking form: Look for consistent spacing of points around a straight line; if the pattern bends or asymptotes, it's nonlinear, and logarithmic or other transformations might linearize it.

Question 9

1. The points show an accelerating increase (curvature), not a constant rate of change. (correct answer)
2. Because the association is positive, a linear model is always appropriate.
3. There are no points exactly on a line, so linear regression cannot be used.
4. One low point proves the relationship is nonlinear.
5. The x-variable is in thousands, which makes a linear model inappropriate.

Explanation: AP Statistics emphasizes analyzing departures from linearity by spotting non-straight patterns in scatterplots, like the accelerating increase here in sales versus advertising. The described scatterplot shows sales increasing slowly at first and then more rapidly, suggesting an upward curve with a non-constant rate of change. Choice A correctly notes this accelerating curvature as the feature making a linear model inappropriate. Distractor choice B claims positive associations always fit linearly, but positivity doesn't ensure a constant slope. Choice C is incorrect because imperfect alignment doesn't preclude linearity if the overall trend is straight. To check form: Imagine fitting a line; if residuals show a systematic curve (e.g., quadratic), it's nonlinear, and polynomial regression or data transformations could be explored.