College Statistics Quiz: Residual Plots

20 questions · exam conditions

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Residual PlotsQuestion 1 of 20

An economist models the price of a house based on its square footage using a simple linear regression. The resulting residual plot shows the absolute values of the residuals tending to increase as the fitted house price increases. This pattern is often described as a 'fan' or 'cone' shape. What is the primary implication of this observation for the model?

The relationship between square footage and house price is non-linear, and a quadratic model would be more appropriate.

The model's predictions are less reliable for more expensive houses than for less expensive ones.

The intercept of the regression model is biased, causing systematic errors across all price ranges.

The model is likely to systematically overestimate the price of smaller houses and underestimate the price of larger houses.

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College Statistics Quiz

College Statistics Quiz: Residual Plots

Practice Residual Plots in College Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Residual Plots, giving you a quick way to practice the rules, question types, and explanations that matter most for College 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.

All questions

Question 1

The relationship between square footage and house price is non-linear, and a quadratic model would be more appropriate.
The model's predictions are less reliable for more expensive houses than for less expensive ones. (correct answer)
The intercept of the regression model is biased, causing systematic errors across all price ranges.
The model is likely to systematically overestimate the price of smaller houses and underestimate the price of larger houses.

Explanation: A fan-shaped residual plot indicates heteroscedasticity, meaning the variance of the residuals is not constant. In this case, the variance of the errors increases as the predicted value increases. This means the model's predictions have more uncertainty (are less reliable) for higher-priced, larger houses.

Question 2

A residual plot for a linear regression model contains one point with an extremely large negative residual. All other residuals are clustered in a narrow band around zero. From the residual plot alone, what can be concluded about this single point?

The point must be removed from the data because it is an error.
The point has high leverage and is therefore influential on the slope of the regression line.
The model's prediction for this point was substantially higher than the actual observed value. (correct answer)
The value of the explanatory variable for this point is larger than for all other points.

Explanation: A residual is

y_{actual} - y_{predicted}

. An extremely large negative residual means this difference is a large negative number, which implies that

y_{predicted}

was much larger than

y_{actual}

. Therefore, the model substantially overestimated the response. The residual plot itself does not provide enough information to determine leverage, influence, or whether the point is an error.

Question 3

A student fits a simple linear model to a set of data and finds that the residual plot shows no discernible pattern; the points appear as a random horizontal band. However, the student is concerned because the correlation coefficient is only $r = 0.30$ . What should the student conclude?

An error was likely made, as a random residual plot should only occur when the correlation is strong.
The low correlation coefficient indicates the linear model is inappropriate, despite the appearance of the residual plot.
The student should try fitting a non-linear model to increase the correlation coefficient.
The linear model is appropriate, but the predictive relationship between the variables is weak. (correct answer)

Explanation: When evaluating linear regression models, you need to understand the distinction between model appropriateness and model strength. The residual plot tells you whether a linear model is appropriate, while the correlation coefficient tells you how strong the relationship is. A random, horizontal band pattern in the residual plot is exactly what you want to see—it indicates that the linear model assumptions are met and that linear regression is the appropriate modeling approach. The residuals show no systematic patterns, curved relationships, or increasing/decreasing variance, confirming that the linear model correctly captures the form of the relationship between variables. The correlation coefficient

r = 0.30

indicates a weak positive linear relationship. While this means the model won't make highly accurate predictions, it doesn't invalidate the appropriateness of using linear regression. Many real-world relationships are legitimately weak but still linear in nature. Answer A is incorrect because residual plots should be random regardless of correlation strength—even weak linear relationships produce random residual patterns when properly modeled. Answer B misunderstands the role of residual plots; they assess model appropriateness, not the correlation coefficient. Answer C suggests an unnecessary step since the residual plot already confirms that linear modeling is appropriate, and switching to non-linear models won't necessarily improve correlation. Remember this key distinction: residual plots diagnose whether your chosen model type fits the data structure, while correlation coefficients measure the strength of that relationship. A good model can have weak predictive power—that's often just the nature of the data, not a modeling error.

Question 4

After fitting a linear model, $\hat{y} = b_0 + b_1x$ , a researcher plots the residuals against a second potential explanatory variable, $z$ , which was not included in the original model. This plot reveals a strong negative linear relationship. What is the most likely implication?

The variable $z$ is strongly correlated with $x$ , a condition known as multicollinearity.
The relationship between $x$ and $y$ is non-linear, and the model needs a quadratic term for $x$ .
The variable $z$ contains additional information that could improve the model's predictions if it were included. (correct answer)
The variance of the errors in the original model is not constant, and a data transformation on $y$ is needed.

Explanation: When residuals from a model show a pattern when plotted against a new variable, it means that the new variable can explain some of the leftover variance that the original model could not. A strong pattern suggests that this new variable,

z

, is a useful predictor and should be considered for inclusion in a multiple regression model to improve its accuracy.

Question 5

A linear regression of weight on height for a sample of adults produces a residual plot with a pronounced fan shape, opening to the right. A biologist suggests transforming the response variable, weight, to improve the model. Which transformation is most likely to be helpful in this situation, and why?

Taking the square of the weight, because this will increase the variance of the errors for taller individuals.
Taking the natural logarithm of the weight, because this will compress larger values more than smaller values, potentially stabilizing the variance. (correct answer)
Standardizing the weight (subtracting the mean and dividing by the standard deviation), as this will center the residuals around zero.
Taking the reciprocal of the weight, because this transformation is the most effective method for correcting non-linear patterns.

Explanation: A fan-shaped residual plot signifies non-constant variance (heteroscedasticity), where the error variance grows with the predicted value. To correct this, a transformation that 'pulls in' the larger values more than the smaller values is often used. The logarithmic transformation is a common and effective choice for this purpose, as it can stabilize the variance. Squaring would make the problem worse. Standardization does not change the fundamental pattern of the residuals. A reciprocal is a strong transformation usually used for different kinds of patterns.

Question 6

A linear regression is performed to predict blood pressure from age. An examination of the residual plot reveals that for individuals between the ages of 20 and 30, nearly all the residuals are positive. What is the most direct interpretation of this finding?

There is more variability in blood pressure among people in their 20s than in other age groups.
The linear model tends to overestimate the blood pressure of people in their 20s.
The relationship between age and blood pressure is weaker for people in their 20s.
The linear model tends to underestimate the blood pressure of people in their 20s. (correct answer)

Explanation: When analyzing residual plots in linear regression, you're looking for patterns that reveal how well your model fits the data. Residuals are the differences between actual observed values and predicted values from your regression line. If nearly all residuals are positive for people aged 20-30, this means the actual blood pressure values are consistently higher than what the linear model predicts for this age group. Since residual = actual value - predicted value, positive residuals indicate the model's predictions are too low. Therefore, answer D is correct: the linear model tends to underestimate the blood pressure of people in their 20s. Let's examine why the other options miss the mark. Option A focuses on variability (spread) rather than systematic bias - having all positive residuals indicates consistent underestimation, not increased variability. Option B gets the direction backward: if residuals are positive, the model is predicting values that are too low, not too high (overestimating would create negative residuals). Option C discusses the strength of the relationship, but systematic bias in one age group doesn't necessarily indicate a weaker correlation - it suggests the linear model isn't capturing the true pattern well for that subset. Remember this key insight: when you see a cluster of residuals with the same sign in a specific range, think "systematic bias." Positive residuals mean underestimation, negative residuals mean overestimation. This pattern often suggests the relationship might not be perfectly linear across all values of the predictor variable.

Question 7

A statistician fits a linear regression model and obtains an R-squared value of 0.22. The corresponding residual plot shows a seemingly random scatter of points with a uniform vertical spread centered on zero. Which statement is the most valid interpretation of these results?

The linear model is appropriate for the data, but the explanatory variable does not account for a large portion of the variability in the response. (correct answer)
The low R-squared value implies that the linear model is a poor fit, and the randomness of the residual plot is therefore misleading.
The random scatter in the residual plot guarantees that predictions from the model will be highly accurate.
There must be an error in the analysis, as a properly fitted model should have both a high R-squared value and a random residual plot.

Explanation: A random residual plot suggests that the assumptions of linearity and constant variance are met, meaning the linear model form is appropriate. However, R-squared measures the proportion of variance explained. A low R-squared (0.22) means the explanatory variable has weak predictive power. These two facts are not contradictory; a model can be correctly specified yet have low predictive power.

Question 8

The residual plot for a linear regression of y on x shows that the residuals are consistently positive for small x-values, negative for medium x-values, and positive again for large x-values. Which of the following is the most likely appearance of the original scatterplot of y versus x with the least-squares regression line superimposed?

The points form a tight, straight line, and the regression line passes directly through them.
The points form an inverted U-shaped curve, and the regression line lies above the points in the middle and below them at both ends.
The points form a U-shaped curve, and the regression line lies below the points in the middle and above them at both ends.
The points form a U-shaped curve, and the regression line lies below the points at the ends and above them in the middle. (correct answer)

Explanation: A positive residual means observed y > predicted y (the point is above the line). A negative residual means observed y < predicted y (the point is below the line). The description matches points that are above the line for small and large x, and below the line for medium x. This pattern occurs when a straight line is fit to a U-shaped (concave up) scatter of points.

Question 9

On a standard residual plot where residuals are plotted against the fitted values ( $\hat{y}$ ), the horizontal line drawn at a residual value of 0 represents which of the following?

The cases where the model's prediction was perfectly accurate. (correct answer)
The mean value of the response variable, $\bar{y}$ .
The mean value of the explanatory variable, $\bar{x}$ .
The set of all possible values for the regression slope.

Explanation: A residual is defined as the observed value (y) minus the predicted value (

\hat{y}

). A residual of 0 means that

y - \hat{y} = 0

, which implies

y = \hat{y}

. Therefore, any point lying on the horizontal line at 0 represents an observation for which the model's prediction was exactly equal to the actual outcome.

Question 10

A researcher fits a simple linear regression model to predict crop yield (in kg/hectare) from the amount of a new fertilizer applied (in kg/hectare). A plot of the residuals versus the fitted values shows a distinct U-shaped pattern. Which of the following is the most appropriate conclusion?

The model systematically overpredicts crop yield for moderate fertilizer amounts and underpredicts it for very low or very high amounts. (correct answer)
The variance of the crop yield is not constant across different fertilizer amounts, indicating a need for data transformation.
The relationship between fertilizer amount and crop yield is weak or non-existent, as evidenced by the clear pattern in the residuals.
The linear model is a good fit because the residual plot reveals a clear, predictable pattern in the errors.

Explanation: A U-shaped residual plot indicates that the linear model is not appropriate and a non-linear relationship exists. The 'U' shape means residuals are positive (underprediction) at the extremes (low and high x-values) and negative (overprediction) in the middle range. Thus, the model overpredicts for moderate amounts and underpredicts for low and high amounts.

Question 11

The linear model is appropriate for the data, but the explanatory variable does not account for a large portion of the variability in the response. (correct answer)
The low R-squared value implies that the linear model is a poor fit, and the randomness of the residual plot is therefore misleading.
The random scatter in the residual plot guarantees that predictions from the model will be highly accurate.
There must be an error in the analysis, as a properly fitted model should have both a high R-squared value and a random residual plot.

Question 12

A linear model is created to predict students' final exam scores from the number of hours they studied. The model's residual plot exhibits a strong 'fan-out' pattern (heteroscedasticity), where the residuals are more spread out for students who studied longer. If this model is used to create a 95% prediction interval for the score of a student who studied for many hours, what can be said about this interval?

The interval will likely be too narrow, understating the true uncertainty in the prediction. (correct answer)
The interval will likely be too wide, overstating the true uncertainty in the prediction.
The center of the prediction interval will be systematically lower than the student's true score.
The prediction interval will be reliable, as heteroscedasticity does not affect prediction intervals.

Explanation: Standard prediction intervals are calculated assuming constant variance (homoscedasticity). They use an estimate of error variance that is an average across all data points. When heteroscedasticity is present (variance increases with x), this average variance will be too small for high-x values. Consequently, the calculated prediction interval for a student who studied many hours will be narrower than it should be, giving a false sense of precision.

Question 13

An analyst examines two residual plots from two different linear regression models. Plot A shows a clear sine-wave pattern. Plot B shows a random cloud of points that gets wider as the fitted values increase. Which statement best distinguishes the problems in Model A and Model B?

Model A violates the linearity assumption, while Model B violates the constant variance assumption. (correct answer)
Model A violates the constant variance assumption, while Model B violates the linearity assumption.
Both models primarily violate the linearity assumption, but in different ways.
Both models primarily violate the assumption that the errors are independent.

Explanation: Any systematic, non-random pattern in the residuals (like a curve or sine wave) suggests the relationship between the variables is not linear. Therefore, Model A violates the linearity assumption. A pattern where the vertical spread of the residuals changes as the fitted values change (like a fan or cone) suggests that the variance of the errors is not constant. Therefore, Model B violates the constant variance (homoscedasticity) assumption.

Question 14

A linear model is used to predict daily ice cream sales from the daily high temperature. The data were collected over 90 consecutive summer days. A plot of the residuals versus the observation day (1 to 90) shows that long stretches of positive residuals are followed by long stretches of negative residuals. What regression assumption is most clearly violated?

Linearity of the relationship between sales and temperature.
Normality of the residuals' distribution.
Independence of the residuals. (correct answer)
Homoscedasticity (constant variance) of the residuals.

Explanation: Plotting residuals against time or observation order is a way to check for independence. A pattern where residuals are not randomly scattered but instead show trends or cycles indicates that the error in one observation is related to the error in another. This pattern, where positive values tend to follow positive values and negative follow negative, is called positive autocorrelation and is a violation of the independence assumption.

Question 15

A marketing analyst suspects that the effect of advertising spending on sales is non-linear, with diminishing returns at higher spending levels. After fitting a simple linear model, which diagnostic plot would be the most direct and effective tool for confirming this suspicion?

A histogram of the sales data to check for skewness in the response variable.
A plot of residuals versus fitted values to detect systematic patterns in the model's errors. (correct answer)
A scatterplot of sales versus advertising spending to visually inspect the relationship.
A normal probability plot of the residuals to check the assumption of normally distributed errors.

Explanation: When you suspect a non-linear relationship after fitting a linear model, you need to examine whether your model is systematically missing the true pattern in the data. The most effective way to detect this is through residual analysis. A residuals versus fitted values plot (option B) is your best diagnostic tool here. If the relationship is truly non-linear with diminishing returns, your linear model will consistently overpredict at low and high spending levels while underpredicting in the middle range. This creates a curved pattern in the residual plot—you'll see negative residuals at the extremes and positive residuals in the middle, forming a distinctive U-shape or inverted U-shape. This systematic pattern directly reveals that your linear model is missing the non-linear structure. Option A is incorrect because a histogram of sales data only shows the distribution of your response variable, not the relationship between variables. Option C, while useful for initial exploration, isn't the most effective after you've already fitted a model—the residual plot provides clearer evidence of model inadequacy by removing the linear trend. Option D checks the normality assumption, which is important for inference but won't help you detect non-linearity in the underlying relationship. Study tip: Remember that residual plots are your primary tool for model validation. When residuals show patterns (curves, trends, or systematic deviations), your model is missing something important about the data structure. Random scatter indicates a well-fitting model, while patterns signal the need for model modifications.

Question 16

An analyst evaluates a residual plot that shows a random cloud of points centered at $y=0$ with no change in vertical spread. However, the density of the points is much greater for fitted values between 10 and 20 than for fitted values between 40 and 50. What is a reasonable inference from this observation?

The linear model is inappropriate because the relationship is curved.
The variance of the errors is not constant, violating the homoscedasticity assumption.
The mean of the residuals is not zero, indicating a systematic bias in the model.
The original dataset likely had a higher concentration of data points in the lower range of the explanatory variable. (correct answer)

Explanation: When analyzing residual plots in regression, you're checking whether the model meets key assumptions. A residual plot shows residuals (actual minus predicted values) on the y-axis and fitted values on the x-axis. The ideal plot shows a random scatter around zero with constant spread. The key insight here is distinguishing between the pattern of residuals and their density. This plot shows good pattern characteristics—random scatter around zero with constant vertical spread—but varying point density across different fitted value ranges. This density variation reflects the original data distribution, not model problems. Option D is correct because when you have more original data points in lower ranges of your explanatory variable, those points get transformed into fitted values that cluster in corresponding ranges. The regression line maps your original x-values to fitted values, so dense regions in your original data create dense regions in the residual plot. Option A is wrong because a curved relationship would show systematic patterns in the residuals (like a U-shape), not just varying density. Option B incorrectly identifies a homoscedasticity violation—the vertical spread remains constant, so variance is consistent. Option C misses that the residuals are centered at zero; varying density doesn't affect the mean. Strategy tip: On residual plot questions, separate pattern problems (which indicate model violations) from density differences (which reflect your original data distribution). Pattern issues like curves, funneling, or systematic deviations from zero indicate model problems, while uneven point density simply mirrors where your original data was concentrated.

Question 17

The model systematically overpredicts crop yield for moderate fertilizer amounts and underpredicts it for very low or very high amounts. (correct answer)
The variance of the crop yield is not constant across different fertilizer amounts, indicating a need for data transformation.
The relationship between fertilizer amount and crop yield is weak or non-existent, as evidenced by the clear pattern in the residuals.
The linear model is a good fit because the residual plot reveals a clear, predictable pattern in the errors.

Question 18

A chemist develops a linear model to predict the reaction rate based on temperature and finds a strong coefficient of determination ( $R^2 = 0.92$ ). However, a plot of the residuals versus temperature shows a clear parabolic (curved) pattern. Which conclusion is most justified?

The high $R^2$ value indicates the linear model is highly effective and suitable for making predictions.
The relationship between temperature and reaction rate is weak, and the high $R^2$ is likely spurious.
Despite the high $R^2$ , the linear model is inappropriate because the relationship is fundamentally non-linear. (correct answer)
The parabolic pattern in the residuals suggests that the variance of the errors is not constant.

Explanation: A residual plot is used to check the assumptions of the linear model. A clear pattern, such as a parabola, indicates that the linearity assumption is violated. Even if the linear model can explain a large portion of the variance (high R-squared), it is not the correct model form. It will produce systematic, predictable errors. The pattern points to a non-linear relationship, not heteroscedasticity (which would be a change in vertical spread).

Question 19

Linearity of the relationship between sales and temperature.
Normality of the residuals' distribution.
Independence of the residuals. (correct answer)
Homoscedasticity (constant variance) of the residuals.

Question 20

A histogram of the sales data to check for skewness in the response variable.
A plot of residuals versus fitted values to detect systematic patterns in the model's errors. (correct answer)
A scatterplot of sales versus advertising spending to visually inspect the relationship.
A normal probability plot of the residuals to check the assumption of normally distributed errors.