Aimathic – Residual plots – Free math Worksheets

Residual plots

Click problems to add them to your worksheet.

Difficulty: 1 – 5

53957012

For a regression relating the number of paint layers to drying time, consider the statement: “A positive residual means the prediction was too high.” Correct the statement.

Hints

- Start with residual $=y-\hat{y}$. - Determine what a positive difference says about $y$ and $\hat{y}$. - Translate that comparison into overprediction or underprediction.

Solution

1. A residual is defined as observed response minus predicted response: $y-\hat{y}$. 2. A positive residual means $y>\hat{y}$, so the observed drying time is above the prediction. 3. Therefore, the model predicted too low, not too high.

Answer

A positive residual means the observed response is greater than the predicted response, so the model predicted too low.

53957112

For a regression relating the number of musicians to setup time, consider the statement: “Residuals should all equal $0$ for a useful model.” Correct the statement.

Hints

- Distinguish exact prediction from an adequate overall model. - Think about natural variation in real observations. - Focus on the pattern and spread of residuals, not whether each one is exactly $0$.

Solution

1. A residual of $0$ means one observation is predicted exactly, but a useful model does not need to predict every observation exactly. 2. Because real data vary, useful models generally have nonzero residuals. 3. The important diagnostic is that the residuals are scattered around $0$ without systematic patterns and with roughly constant spread.

Answer

A useful model can have nonzero residuals. The residuals should be scattered around $0$ without a systematic pattern and with roughly constant spread.

53957212

For a regression relating distance from a lamp to light intensity, consider the statement: “A residual plot with a U-shape confirms linearity because it is centered at $0$.” Correct the statement.

Hints

- Separate the residual plot's center from its shape. - Ask whether the residuals are random or form a systematic curve. - Averages can be near $0$ even when the model misses important structure.

Solution

1. Residuals from a suitable linear model should be randomly scattered around $0$. 2. Being centered at $0$ is not sufficient if the residuals form a systematic shape. 3. A U-shaped pattern indicates curvature that the fitted line does not capture, so it is evidence against linearity.

Answer

The statement is incorrect. A U-shaped residual pattern indicates unmodeled curvature, so the linear form is inappropriate even if the residuals are centered at $0$ overall.

53957312

For a regression relating hours after sunrise to shadow length, consider the statement: “The point with the largest residual must also have the largest $x$-value.” Correct the statement.

Hints

- Distinguish horizontal position from vertical distance to the fitted line. - The largest $x$-value concerns leverage. - The largest residual concerns the greatest observed-minus-predicted difference.

Solution

1. A residual measures vertical prediction error, $y-\hat{y}$. 2. The largest $x$-value identifies the observation farthest to the right, which concerns leverage rather than residual size. 3. An observation at any $x$-value can have the largest positive or negative residual.

Answer

The statement is incorrect. Residual size measures vertical prediction error and is not determined by the size of $x$; any observation can have the largest residual.

54807312

A regression predicts fuel economy $y$, in miles per gallon, from engine displacement $x$, in liters. For one vehicle, the residual is $-2.4$. What units should be attached to the residual, and what does the value mean in context?

Hints

- Identify which variable appears in the observed-minus-predicted difference. - Differences keep the units of the quantities being subtracted. - Use the sign to decide whether the observation is above or below the fitted line.

Solution

1. A residual is $y-\hat y$, so it is a difference between two response values. 2. Therefore, the residual has the response units: miles per gallon. 3. A residual of $-2.4$ means the observed fuel economy is $2.4$ miles per gallon lower than the model predicted for that engine displacement.

Answer

The residual is $-2.4$ miles per gallon. The vehicle’s observed fuel economy is $2.4$ miles per gallon below its predicted value.

54809912

For one observation in a linear regression, $x=12$, the observed response is $y=30$, and the fitted model gives $\hat y=27.5$. What point represents this observation on a residual plot with $x$ on the horizontal axis? Interpret its vertical coordinate.

Hints

- Find the vertical prediction error before locating the point on the residual plot. - Keep the original explanatory value as the horizontal coordinate. - Use the sign of the residual to decide whether the model overpredicts or underpredicts.

Solution

1. The residual is $e=y-\hat y=30-27.5=2.5$. 2. A residual plot against the explanatory variable uses the original $x$-value and the residual, so the point is $(12, 2.5)$. 3. The positive residual means the observed response is $2.5$ units above the model's prediction at $x=12$.

Answer

The residual-plot point is $(12, 2.5)$. The model underpredicts this observation by $2.5$ response units.

54811212

Suppose the original sloped regression line is drawn across a residual graph as a reference. What reference line should be used on a residual plot instead, and what does it represent?

Hints

- Think about what residual value corresponds to a perfect prediction. - Residuals use a new vertical scale centered on prediction error. - The appropriate baseline should have the same meaning at every horizontal position.

Solution

1. A residual measures observed response minus fitted response, so a residual of $0$ means the observation lies exactly on the fitted regression line. 2. The natural reference in residual coordinates is therefore the horizontal line $e=0$. 3. The original regression line is expressed in response coordinates, not residual coordinates, so drawing it on the residual plot does not provide the appropriate baseline.

Answer

Use the horizontal line at residual $0$. It represents observations for which the fitted model predicts the response exactly.

53955112

A linear model predicting accuracy score from practice time, in hours, is $\hat y = 15.0 + 1.4x$. For an observation with $x = 3$, the observed accuracy score is $y = 17.1$. Find and interpret the residual.

Hints

- Substitute $x=3$ into the model to find $\hat y$. - Use residual $=y-\hat y$. - A negative residual means the observed response is below the prediction.

Solution

1. Find the predicted accuracy score: $\hat y = 15.0 + 1.4(3) = 19.2$. 2. Compute observed minus predicted: $y-\hat y = 17.1-19.2 = -2.1$. 3. The negative residual means the observed accuracy score is $2.1$ points below the model’s prediction, so the model overpredicts by $2.1$ points.

Answer

The residual is $-2.1$. The observed accuracy score is $2.1$ points below the prediction, so the model overpredicts by $2.1$ points.

53955212

A linear model predicting temperature, in degrees Fahrenheit, from distance from a heater, in feet, is $\hat y = 17.0 - 1.6x$. For an observation with $x = 4$, the observed temperature is $y = 12.1$. Find and interpret the residual.

Hints

- Substitute $x=4$ into the model to find the predicted temperature. - Compute residual $=y-\hat y$. - A positive residual means the observed response is above the prediction.

Solution

1. Find the predicted temperature: $\hat y = 17.0 - 1.6(4) = 10.6$. 2. Compute observed minus predicted: $y-\hat y = 12.1-10.6 = 1.5$. 3. The positive residual means the observed temperature is $1.5$ degrees Fahrenheit above the model’s prediction, so the model underpredicts by $1.5$ degrees Fahrenheit.

Answer

The residual is $1.5$. The observed temperature is $1.5$ degrees Fahrenheit above the prediction, so the model underpredicts by $1.5$ degrees Fahrenheit.

53955312

A linear model predicting shipping cost, in dollars, from package weight, in pounds, is $\hat y = 19.0 + 1.8x$. For a package with $x = 5$, the observed shipping cost is $y = 28.0$. Find and interpret the residual.

Hints

- Substitute $x=5$ into the regression equation. - Use residual $=y-\hat y$. - Interpret what it means when the observed and predicted responses are equal.

Solution

1. Find the predicted shipping cost: $\hat y = 19.0 + 1.8(5) = 28.0$. 2. Compute observed minus predicted: $y-\hat y = 28.0-28.0 = 0.0$. 3. A residual of $0.0$ means the observed shipping cost equals the model’s prediction exactly.

Answer

The residual is $0.0$. The observed shipping cost exactly matches the model’s prediction.

53955412

For a regression model predicting car speed, in miles per hour, from ramp height, an observation has predicted response $\hat y = 20.0$ and residual $4.2$. Find the observed response and state whether the model predicted too high or too low.

Hints

- Rearrange residual $=y-\hat y$ to solve for $y$. - Add the residual to the predicted response. - A positive residual means the observation is above the fitted value.

Solution

1. Use residual $=y-\hat y$, so $y=\hat y+\text{residual}$. 2. The observed speed is $y=20.0+4.2=24.2$ miles per hour. 3. The residual is positive, so the observed speed is above the prediction. Therefore, the model predicted too low.

Answer

The observed response is $y=24.2$ miles per hour, and the model predicted too low.

53955512

For a regression model predicting reading time, in minutes, from the number of pages, an observation has predicted response $\hat y = 23.0$ and residual $-3.5$. Find the observed response and state whether the model predicted too high or too low.

Hints

- Rearrange residual $=y-\hat y$ to solve for the observed response. - Add the negative residual to the predicted value. - A negative residual means the observation is below the fitted value.

Solution

1. Use residual $=y-\hat y$, so $y=\hat y+\text{residual}$. 2. The observed reading time is $y=23.0+(-3.5)=19.5$ minutes. 3. The residual is negative, so the observed reading time is below the prediction. Therefore, the model predicted too high.

Answer

The observed response is $y=19.5$ minutes, and the model predicted too high.

53955612

For a regression model predicting resale price, in thousands of dollars, from vehicle age, an observation has response $y = 41.0$ and residual $3.3$. Find the predicted response.

Hints

- Start with residual $=y-\hat y$. - Isolate $\hat y$ before substituting the values. - Subtract the positive residual from the observed response.

Solution

1. Rearrange residual $=y-\hat y$ to $\hat y=y-\text{residual}$. 2. Compute $\hat y=41.0-3.3=37.7$. 3. The model predicted a resale price of $37.7$ thousand dollars.

Answer

$\hat y = 37.7$, or $\$37{,}700$.

53955712

For a regression model predicting air pressure, in kilopascals, from elevation, an observation has response $y = 43.0$ and residual $-4.6$. Find the predicted response.

Hints

- Start with residual $=y-\hat y$. - Isolate $\hat y$ before substituting. - Subtracting a negative residual increases the predicted value.

Solution

1. Rearrange residual $=y-\hat y$ to $\hat y=y-\text{residual}$. 2. Compute $\hat y=43.0-(-4.6)=47.6$. 3. The model predicted an air pressure of $47.6$ kilopascals.

Answer

$\hat y = 47.6$ kilopascals.

53955812

Use the residual plot for a linear regression predicting wait time from crowd size. What do the pattern and spread indicate about the model?

Hints

- Look for a systematic curve or trend in the residuals. - Compare the vertical spread at small and large crowd sizes. - State only the model features that a residual plot can assess.

Solution

1. Random scatter around $0$ indicates that no systematic curved pattern remains. 2. Similar vertical spread across the plot supports the constant-variance condition. 3. Therefore, the residual plot supports using a linear model with approximately constant residual variance. It does not by itself verify every regression condition.

Answer

The residual plot supports a linear model with approximately constant residual variance because it shows random scatter around $0$ with no change in spread.

53955912

Use the residual plot for a linear regression predicting dissolving time from water temperature. What does the pattern indicate about the model?

Hints

- Track how the residual signs change from low to middle to high temperatures. - A suitable linear model should leave no systematic shape in the residuals. - Connect a U-shaped pattern with curvature in the original relationship.

Solution

1. The residuals form a curved, U-shaped pattern rather than random scatter around $0$. 2. Positive residuals at the ends mean the line tends to underpredict there, while negative residuals in the middle mean it tends to overpredict there. 3. This systematic pattern indicates that the relationship is nonlinear and that the linear model is missing curvature.

Answer

The U-shaped residual pattern indicates that the relationship is nonlinear and the fitted line is missing curvature.

53956012

Use the residual plot for a linear regression predicting bowl diameter from clay mass. What does the pattern indicate about the model?

Hints

- Compare the vertical spread of residuals at small and large clay masses. - A suitable constant-variance model should have roughly equal spread across the plot. - Name the condition violated by a widening fan shape.

Solution

1. The residuals form a fan shape because their vertical spread increases with clay mass. 2. This pattern indicates that the residual variance is not constant across values of the explanatory variable. 3. Therefore, the constant-variance condition for the linear regression model is violated.

Answer

The fan-shaped residual pattern indicates nonconstant residual variance; prediction errors become more variable as clay mass increases.

53956112

Use the residual plot for a linear regression predicting fuel used from miles traveled. What does the plot indicate about the model?

Hints

- Use the residual's sign to compare the observed fuel use with the prediction. - Distinguish a large residual from high leverage. - Consider what additional information would be needed to judge influence on the fitted line.

Solution

1. The isolated large positive residual identifies an observation whose fuel use is much greater than the model predicts. 2. The observation is an outlier in the response direction and should be checked for a recording error or a relevant unusual circumstance. 3. The residual plot alone does not show whether the point has high leverage or strongly changes the fitted line; that depends on its explanatory value.

Answer

The plot shows an observation with an unusually large positive residual. It should be investigated, but the residual plot alone does not determine whether it is influential.

53956312

Use the residual plot for a linear regression predicting sailing time from wind speed. What does the pattern indicate about the model?

Hints

- A suitable linear model should leave residuals randomly scattered around $0$. - Trace the repeated changes in residual sign as wind speed increases. - A systematic S-shape suggests that one straight line cannot capture the relationship.

Solution

1. The residuals follow a systematic S-shaped pattern rather than random scatter around $0$. 2. This pattern indicates that the relationship changes curvature across the range of wind speeds. 3. Therefore, the fitted linear model does not adequately represent the relationship; a nonlinear model should be considered.

Answer

The S-shaped residual pattern indicates unmodeled nonlinearity, so the fitted linear model is inadequate.

53957412

A residual table for a fitted least-squares regression line with an intercept, relating the number of proofreads to the number of typos remaining, lists residuals $-2.4$, $1.1$, $0.8$, $-0.5$, and one missing value. Use the residual-sum property to find the missing residual.

Hints

- Recall the sum-of-residuals property for a least-squares line with an intercept. - Add the four known residuals, keeping their signs. - Choose the missing value that makes the total $0$.

Solution

1. For a least-squares line with an intercept, the residuals sum to $0$. 2. Let the missing residual be $e$: $-2.4+1.1+0.8-0.5+e=0$. 3. The known residuals sum to $-1.0$, so $-1.0+e=0$. 4. Therefore, $e=1.0$.

Answer

The missing residual is $1.0$.

53957512

The residuals from a regression predicting top speed from kayak length are $-1.2$, $0.7$, $0.4$, $-0.3$, and $0.4$. Find their mean and explain whether it is consistent with a least-squares line that includes an intercept.

Hints

- Add the residuals with their signs before dividing. - The mean equals the residual sum divided by $5$. - Compare the result with the residual-mean property of a least-squares fit with an intercept.

Solution

1. Add the residuals: $-1.2+0.7+0.4-0.3+0.4=0.0$. 2. Divide by the number of residuals: $\frac{0.0}{5}=0.0$. 3. A least-squares line with an intercept has residuals with mean $0$, so this result is consistent with that property.

Answer

The mean residual is $0.0$, which is consistent with a least-squares regression line that includes an intercept.

53957612

Residual plots A and B compare two models for predicting travel time from snow depth. They use the same observations and response units. Which model gives more precise predictions? Explain.

Hints

- Check each plot for a systematic pattern before comparing precision. - Compare the vertical residual spreads using the common scale. - Smaller residual magnitudes mean observations are generally closer to predictions.

Solution

1. Neither plot shows a systematic pattern, so both model forms are plausible. 2. Model A’s residuals have a smaller vertical spread than Model B’s residuals. 3. Therefore, Model A’s predictions are typically closer to the observed travel times and are more precise.

Answer

Model A gives more precise predictions because its residuals have the smaller spread.

54795812

A regression model predicts a response of $46.5$ for one observation. The recorded response was originally $48.0$, but a data check shows that the correct observed value is $44.0$. Find the original residual and the corrected residual. By how much did the residual change, and does the corrected observation lie above or below the fitted line?

Hints

- Keep the model prediction fixed while updating the observed response. - Compute the residual the same way before and after the correction. - Use the sign of the corrected residual to locate the observation relative to the fitted line.

Solution

1. The original residual is $48.0-46.5=1.5$. 2. The corrected residual is $44.0-46.5=-2.5$. 3. The residual changed by $-2.5-1.5=-4.0$. 4. The corrected residual is negative, so the corrected observation lies below the fitted line and the model overpredicts it by $2.5$.

Answer

Original residual: $1.5$. Corrected residual: $-2.5$. The residual decreased by $4.0$, and the corrected observation lies below the fitted line.

54797112

Consider the statement: “A residual plot is only valid when residuals are plotted against the original explanatory variable. Plotting residuals against the predicted responses is wrong.” Evaluate the statement.

Hints

- Focus on the purpose of a residual plot rather than one specific axis choice. - Ask what quantities can organize the fitted observations while leaving residuals on the vertical axis. - Decide whether the alternative plot can still reveal systematic residual patterns.

Solution

1. A residual plot can use either the explanatory-variable values or the fitted response values on the horizontal axis. 2. In either version, the residuals are displayed vertically so patterns such as curvature, changing spread, or unusual residuals can be investigated. 3. Therefore, plotting residuals against predicted responses is a valid residual-plot form.

Answer

The statement is incorrect. Residuals may be plotted against either the explanatory variable or the predicted response values when assessing a regression model.

54797912

A fitted model has residuals $-10$, $-10$, $10$, and $10$. Consider the claim: “The mean residual is $0$, so the model must make very accurate predictions.” Evaluate the reasoning.

Hints

- Compare the average signed error with the sizes of the individual errors. - Ask what happens when positive and negative residuals cancel. - Model accuracy depends on more than the center of the residuals.

Solution

1. The residuals sum to $0$, so their mean is $0$. 2. However, every prediction misses its observed response by $10$ units in absolute value. 3. A mean residual of $0$ can result from positive and negative errors canceling, so it does not show that individual prediction errors are small.

Answer

The reasoning is incorrect. The residual mean is $0$, but every prediction is off by $10$ units, so zero mean residual does not imply accurate predictions.

54798612

Panels a) and b) display exactly the same residuals for the same fitted model, but the vertical-axis ranges are different. Consider the claim: “Model b) is better because its residuals look closer to $0$.” Evaluate the claim.

Hints

- Compare the numerical residual values, not only their apparent distance from the horizontal axis. - Check whether the two panels use the same vertical scale. - Rescaling a graph changes appearance, not the underlying model errors.

Solution

1. The plotted residual coordinates are identical in both panels. 2. Panel b) uses a much wider vertical range, so the same residuals appear visually compressed toward $0$. 3. Changing the graph scale does not change residual size or model fit, so neither panel represents a better model.

Answer

The claim is incorrect. The residuals are identical; panel b) only makes them look smaller by using a wider vertical scale.

54798912

A regression model predicts service time in minutes. For one visit, the observed time is $18.5$ minutes and the predicted time is $16.0$ minutes. Every observed and predicted service time is then converted from minutes to seconds, and residuals are plotted against the same unchanged explanatory variable as before. Find this visit's residual in minutes and in seconds. Then describe how converting the response units affects the overall residual plot and the evidence about whether a linear model is appropriate.

Hints

- Start with the definition of a residual using observed and predicted responses. - Think about what happens to a difference when both response values are multiplied by the same positive conversion factor. - Keep track of which axis changes units and which axis remains unchanged.

Solution

1. In minutes, the residual is $18.5-16.0=2.5$ minutes. 2. Converting both values to seconds multiplies their difference by $60$, so the residual is $2.5\cdot60=150$ seconds. 3. Every residual is multiplied by the same positive constant while the horizontal explanatory-variable values are unchanged. The residual plot is stretched vertically by a factor of $60$, but the signs, relative positions, curvature, clustering, and spread pattern are otherwise unchanged. 4. Therefore, changing from minutes to seconds does not change the evidence about whether the linear form is appropriate.

Answer

The residual is $2.5$ minutes, or $150$ seconds. With residuals plotted against the unchanged explanatory variable, converting to seconds stretches the plot vertically by a factor of $60$ but does not change its diagnostic pattern or the conclusion about linear-model appropriateness.

54800712

Consider the statement: “The residual for each point is the shortest distance from the point to the fitted regression line, measured perpendicular to the line.” Correct the statement and explain how a residual is represented on the scatterplot.

Hints

- Keep the explanatory-variable value fixed when comparing observed and predicted responses. - Identify which coordinate differs between $y_i$ and $\hat{y}_i$. - Think about the units in which a residual is measured.

Solution

1. For an observation at explanatory value $x_i$, the fitted line gives the predicted response $\hat{y}_i$ at that same $x_i$. 2. The residual is $y_i-\hat{y}_i$, which is a vertical difference in response units. 3. Therefore, a residual is represented by the signed vertical separation between the observed point and the fitted line, not by the shortest perpendicular distance to the line.

Answer

A residual is the signed vertical difference $y_i-\hat{y}_i$ at the observation's fixed $x_i$. It is not the perpendicular distance from the point to the regression line.

54801612

Consider the statement: “Because the residuals from a least-squares regression line with an intercept sum to $0$, there must be the same number of positive residuals as negative residuals.” Evaluate the statement. Use the residuals $-8, 2, 2, 2, 2$ as part of your explanation.

Hints

- Add the example residuals and count their signs separately. - Distinguish a balance of signed values from a balance of counts. - A few larger residuals can offset several smaller residuals of the opposite sign.

Solution

1. The residuals $-8, 2, 2, 2, 2$ sum to $0$. 2. This set has four positive residuals and only one negative residual, so a zero residual sum does not require equal counts on the two sides of $0$. 3. The least-squares property balances signed residual magnitudes in total, not the number of positive and negative residuals.

Answer

The statement is incorrect. The residuals $-8, 2, 2, 2, 2$ sum to $0$ even though four are positive and one is negative. A zero residual sum does not imply equal numbers above and below the fitted line.

54803112

A residual plot for an observational data set shows every residual exactly on the horizontal line at $0$. What does this tell you about the fitted linear model for the observed data? Does it prove that changing the explanatory variable causes the response to change? Explain.

Hints

- Translate a residual of $0$ back into a statement about observed and fitted responses. - Separate goodness of fit from the design of the study. - Ask whether a graph alone can establish cause and effect.

Solution

1. Residuals of $0$ for every observation mean every observed response equals its fitted response. 2. Therefore, all observed points lie exactly on the fitted regression line, so the model has a perfect in-sample linear fit and a sum of squared residuals of $0$. 3. A perfect fit describes the observed association but does not establish a cause-and-effect relationship. 4. Causal conclusions depend on the study design and control of alternative explanations, not on residual size alone.

Answer

The plot shows a perfect linear fit to the observed data: every residual is $0$. It does not prove causation; a perfect observational association can still arise without the explanatory variable causing the response.

54804012

A least-squares regression has $\bar{x}=10$ and $\bar{y}=30$. One observation is $(10, 34)$. Consider the statement: “Because the least-squares line passes through $(\bar{x}, \bar{y})$, any observation with $x=\bar{x}$ must have residual $0$.” Evaluate the statement and find the residual of the given observation.

Hints

- Use the point through which the fitted line is guaranteed to pass. - Distinguish the fitted response at $\bar{x}$ from an individual observed response at the same $x$. - Compute the residual as observed minus predicted.

Solution

1. The least-squares line passes through $(10, 30)$, so the predicted response at $x=10$ is $\hat{y}=30$. 2. The observed response for the given point is $34$, not $30$. 3. Its residual is $34-30=4$. 4. The line passing through the point of sample means does not require every observation with $x=\bar{x}$ to have response $\bar{y}$.

Answer

The statement is incorrect. At $x=10$, the fitted value is $30$, so the observation $(10, 34)$ has residual $4$.

54806612

In a regression analysis, one observation has residual $3.6$. Consider the statement: “That means the response was measured with an error of $3.6$ units.” Evaluate the statement.

Hints

- Start from the definition of a residual. - Separate prediction error from measurement error. - Think of several reasons an observation might not lie exactly on a fitted regression line.

Solution

1. A residual is the difference between an observed response and the response predicted by the fitted model at that observation's explanatory value. 2. A nonzero residual can arise from natural variation, omitted variables, imperfect model form, measurement error, or other sources. 3. Therefore, the residual magnitude cannot be identified automatically as measurement error.

Answer

The statement is incorrect. A residual of $3.6$ means the observed response is $3.6$ units above the fitted response. It does not by itself show that the measurement was wrong by $3.6$ units.

54808112

The residual plot comes from a fitted linear regression. Consider the statement: “The pattern proves that the true relationship is quadratic.” Evaluate the statement.

Hints

- Identify what systematic feature appears in the residuals. - Separate what the residual plot rules out from what it uniquely identifies. - More than one nonlinear model can create a similar residual pattern.

Solution

1. The residual plot shows a clear U-shaped pattern, so the linear model leaves systematic curvature unexplained. 2. Therefore, the residual plot provides evidence that a straight-line model is not appropriate over the displayed range. 3. The residual plot alone does not uniquely identify the true functional form; several nonlinear relationships can produce curved residual patterns. 4. Additional modeling and diagnostics are needed before selecting a specific alternative form.

Answer

The U-shaped pattern is evidence against the linear model, but it does not prove that the true relationship is quadratic. It shows unexplained curvature, not a unique nonlinear formula.

54808912

The residual plot comes from a linear model whose explanatory variable can take only the integer values $1, 2, 3, 4, 5$. Consider the claim: “The vertical stacks are a pattern, so the linear model must be inappropriate.” Evaluate the claim.

Hints

- Ask what feature of the explanatory variable creates repeated horizontal positions. - Compare the centers and spreads of the residuals across the stacks. - Separate geometry caused by repeated $x$-values from a systematic residual trend.

Solution

1. Vertical stacks are expected when many observations share the same discrete explanatory value. 2. The relevant diagnostic question is whether the residual centers or spreads change systematically across the explanatory values. 3. In the displayed plot, every stack has residuals above and below zero with similar spread, and there is no systematic change across the five stacks. 4. The stacking itself is therefore not evidence against the linear model.

Answer

The claim is not justified. The stacks result from repeated integer $x$-values; because the residuals remain distributed around zero with similar spread, the stacking alone does not indicate a poor linear model.

54811012

A residual plot contains residuals between about $-4$ and $4$, except for one observation with residual $12$. A presentation sets the vertical axis from $-5$ to $5$, so the residual of $12$ is not visible. Why is this residual plot misleading for evaluating the regression model?

Hints

- Check whether every observation used in the regression is actually represented in the diagnostic display. - Consider how omitting the largest prediction error changes the viewer's impression of the fit. - Diagnostic graphs should reveal unusual observations rather than conceal them through axis choices.

Solution

1. The residual of $12$ is part of the fitted data and is much larger than the other residuals. 2. Restricting the axis so that the point disappears hides an unusual prediction error that may need investigation. 3. Model diagnostics should display all relevant residuals or explicitly indicate values beyond the plotted range.

Answer

The plot is misleading because it hides the largest residual. That omission can make the fit look more regular than it is and prevents the viewer from evaluating an important unusual observation.

54813312

A regression program keeps fitted values to full precision, but suppose every fitted value is rounded to the nearest whole number before residuals are computed and plotted. Why can this create a misleading diagnostic plot?

Hints

- A residual is defined using the model's actual fitted value. - Ask whether rounding occurs before or after the prediction error is calculated. - Small artificial changes repeated across many observations can alter the appearance of a diagnostic plot.

Solution

1. A residual should be computed from the fitted value produced by the regression model, not from an intentionally rounded approximation. 2. Rounding the fitted values changes the residuals by the rounding errors. 3. Those artificial changes can create repeated levels, hide small residual differences, or introduce visual structure that is not present in the model's actual residuals. 4. Diagnostic residuals should therefore be computed from the unrounded fitted values, with rounding used only for final display when appropriate.

Answer

Rounding fitted values before subtraction changes the residuals themselves and can create artificial patterns. The residual plot should be based on the model's unrounded fitted values.

54816012

The histogram of residuals from a linear regression is roughly symmetric and centered near $0$. Why is the histogram alone insufficient to establish an appropriate linear form and constant residual spread?

Hints

- Identify what information a histogram keeps and what paired information it discards. - Model form and equal spread concern how residuals behave across explanatory or fitted values. - An overall distribution can look regular even when residuals are arranged systematically along the regression.

Solution

1. A histogram shows the overall distribution of residual values but discards the explanatory values and fitted values associated with those residuals. 2. Residuals can have a symmetric overall distribution while still showing curvature, changing spread, or another systematic pattern across $x$. 3. Residual plots against $x$ or fitted values are also needed to evaluate linear form and constant variance.

Answer

The histogram cannot show where residuals occur across the predictor range. Symmetric residuals can still have curvature or nonconstant spread, so residual-versus-$x$ or residual-versus-fitted plots are also needed.

53956212

Use the residual plot for a linear regression predicting errors remaining from the number of editing passes. Blue dots represent novice editors, and orange crosses represent experienced editors. What does the pattern indicate about the model?

Hints

- Interpret the sign of the residuals separately for each experience group. - Ask whether the model makes errors in the same direction within a group. - A group-based pattern can indicate an omitted categorical variable or an interaction.

Solution

1. Positive residuals for novice editors mean the model tends to underpredict their remaining errors. 2. Negative residuals for experienced editors mean the model tends to overpredict their remaining errors. 3. The systematic difference by experience group suggests that editor experience is an omitted explanatory variable or that the relationship differs between the groups.

Answer

The model shows systematic bias by editor experience. Experience may be an omitted variable, or separate group relationships may be needed.

53956412

Use the residual plot for an ordinary least-squares regression with an intercept, predicting sound level from distance from a stage. What does the plot indicate?

Hints

- Recall the sum-of-residuals property for least-squares regression with an intercept. - Translate a center near $4$ into a statement about the mean residual. - Decide whether that mean can occur for the stated fitted model.

Solution

1. For an ordinary least-squares regression with an intercept, the residuals must sum to $0$, so their mean is $0$. 2. Residuals centered near $4$ would have a positive mean rather than $0$. 3. Therefore, the displayed values are not the residuals from the stated least-squares fit. The residuals may have been computed incorrectly, or the plotted predictions came from a different line.

Answer

The plot is inconsistent with an ordinary least-squares fit that includes an intercept. Such residuals must be centered at $0$, so the residuals or predictions were computed incorrectly or came from another model.

53956512

Use the residual plot for a linear regression predicting the number of boxes packed from the number of volunteers. What do the two far-right observations indicate?

Hints

- Use horizontal position to assess leverage and vertical distance from $0$ to assess residual size. - A point can be especially influential when both quantities are large. - Opposite residual signs do not guarantee that the points are harmless.

Solution

1. The two observations have extreme explanatory values, so they have high leverage. 2. Their large residuals show that neither observation is well predicted by the fitted line. 3. Points with both high leverage and large residuals can be influential, even though their opposite signs may partly offset each other in the fitted line. 4. Both observations should be investigated, and the model should be refitted with and without them to assess influence.

Answer

The two far-right observations have high leverage and large residuals, so they may be influential. Both should be investigated and their effect on the fitted model assessed.

53956612

Two models for predicting crispness score from hours since baking have the residuals below. The two models have the same set of residual magnitudes, so neither can be preferred by residual size alone. <table> <thead><tr><th>$x$</th><th>Model A residual</th><th>Model B residual</th></tr></thead> <tbody> <tr><td>1</td><td>-2.2</td><td>1.2</td></tr> <tr><td>2</td><td>1.2</td><td>0.8</td></tr> <tr><td>3</td><td>0.8</td><td>0.2</td></tr> <tr><td>4</td><td>-0.2</td><td>-2.2</td></tr> <tr><td>5</td><td>2.2</td><td>-2.2</td></tr> <tr><td>6</td><td>-1.2</td><td>0.2</td></tr> <tr><td>7</td><td>-0.8</td><td>0.8</td></tr> <tr><td>8</td><td>0.2</td><td>1.2</td></tr> </tbody> </table> Which model is more appropriate based on the residual pattern? Explain what feature of the residual sequence determines your choice.

Hints

- Read the signs and sizes in order from the smallest to the largest $x$-value. - Ignore overall residual magnitude as a deciding factor because it is matched between the models. - Look for a sustained shape such as positive residuals at both ends and negative residuals in the middle.

Solution

1. Model A's residuals move above and below $0$ without a sustained shape as $x$ increases. 2. Model B's residuals are positive at low and high $x$-values and strongly negative in the middle, producing a U-shaped pattern. 3. Because the residual magnitudes are the same for the two models, residual size and SSE cannot distinguish them. 4. Model A is more appropriate because its residuals are patternless, while Model B leaves systematic curvature unexplained.

Answer

Model A. Its residuals show no systematic shape, whereas Model B has a U-shaped residual pattern. The models have the same residual magnitudes, so the choice must come from the pattern rather than residual size.

53956712

Two models for predicting visit duration from the number of exhibits visited have the residuals below. The models use exactly the same residual values, but in different orders. <table> <thead><tr><th>$x$</th><th>Model A residual</th><th>Model B residual</th></tr></thead> <tbody> <tr><td>1</td><td>-0.6</td><td>-1.4</td></tr> <tr><td>2</td><td>1.4</td><td>-1.0</td></tr> <tr><td>3</td><td>-1.0</td><td>-0.6</td></tr> <tr><td>4</td><td>0.2</td><td>-0.2</td></tr> <tr><td>5</td><td>1.0</td><td>0.2</td></tr> <tr><td>6</td><td>-1.4</td><td>0.6</td></tr> <tr><td>7</td><td>0.6</td><td>1.0</td></tr> <tr><td>8</td><td>-0.2</td><td>1.4</td></tr> </tbody> </table> Which model is more appropriate based on the residual pattern? Explain why comparing residual size cannot answer the question.

Hints

- Follow each residual column from top to bottom as $x$ increases. - Because the same residual values appear in both columns, focus on their order rather than their magnitudes. - Ask whether one column shows a persistent increase or decrease across the full range.

Solution

1. Model A's residuals alternate irregularly above and below $0$ without an overall trend as $x$ increases. 2. Model B's residuals rise steadily from negative to positive, showing a strong remaining linear trend. 3. The models contain exactly the same residual values, so they have the same residual magnitudes and the same SSE. 4. Model A is more appropriate because its residual sequence is patternless, while Model B shows systematic underprediction at low $x$ and overprediction at high $x$.

Answer

Model A. Model B's residuals increase systematically from negative to positive, while Model A's residuals are irregular around $0$. Because both models use the same residual values, residual size and SSE are identical and cannot determine the choice.

53956812

Two models for predicting flavor score from steeping time have the residuals below. The models use exactly the same residual values, but in different orders. <table> <thead><tr><th>$x$</th><th>Model A residual</th><th>Model B residual</th></tr></thead> <tbody> <tr><td>1</td><td>-1.2</td><td>0.8</td></tr> <tr><td>2</td><td>1.2</td><td>-1.2</td></tr> <tr><td>3</td><td>-0.8</td><td>0.4</td></tr> <tr><td>4</td><td>0.8</td><td>1.0</td></tr> <tr><td>5</td><td>-0.4</td><td>-0.8</td></tr> <tr><td>6</td><td>0.4</td><td>-0.4</td></tr> <tr><td>7</td><td>-1.0</td><td>1.2</td></tr> <tr><td>8</td><td>1.0</td><td>-1.0</td></tr> </tbody> </table> Which model is more appropriate based on the residual pattern? Explain what makes the less appropriate model systematic rather than random.

Hints

- Ignore residual magnitude as a deciding factor because both models contain the same values. - Read only the sequence of signs first. - Decide whether either model repeats a simple pattern at every successive $x$-value.

Solution

1. Model A alternates negative, positive, negative, positive at every successive $x$-value. That exact repeating sign pattern is systematic rather than random scatter. 2. Model B uses the same residual values but places them in an irregular sequence above and below $0$. 3. Because both models use the same residual values, they have the same residual magnitudes and the same SSE. 4. Model B is more appropriate because its residuals are patternless, while Model A leaves a perfectly repeating structure.

Answer

Model B. Model A alternates sign at every successive $x$-value, which is a systematic repeating pattern. Model B uses the same residual values in an irregular order, so residual size cannot provide a shortcut to the choice.

53956912

The model $\hat y = 4.0 + 1.6x$ predicts arrival delay, in minutes, from the number of route changes. <table> <thead><tr><th>$x$</th><th>Observed $y$</th></tr></thead> <tbody> <tr><td>1</td><td>5.20</td></tr> <tr><td>2</td><td>7.40</td></tr> <tr><td>3</td><td>9.35</td></tr> <tr><td>4</td><td>10.15</td></tr> <tr><td>5</td><td>11.90</td></tr> </tbody> </table> Calculate $y-\hat y$ for each observation and complete the residual table.

Hints

- Substitute each $x$-value into $\hat y=4.0+1.6x$. - For each row, compute residual $=y-\hat y$. - Preserve the sign: observations below the fitted line have negative residuals.

Solution

1. At $x=1$, $\hat y=4.0+1.6(1)=5.60$, so the residual is $5.20-5.60=-0.40$. 2. At $x=2$, $\hat y=4.0+1.6(2)=7.20$, so the residual is $7.40-7.20=0.20$. 3. At $x=3$, $\hat y=4.0+1.6(3)=8.80$, so the residual is $9.35-8.80=0.55$. 4. At $x=4$, $\hat y=4.0+1.6(4)=10.40$, so the residual is $10.15-10.40=-0.25$. 5. At $x=5$, $\hat y=4.0+1.6(5)=12.00$, so the residual is $11.90-12.00=-0.10$.

Answer

<table> <thead><tr><th>$x$</th><th>Residual</th></tr></thead> <tbody> <tr><td>1</td><td>-0.40</td></tr> <tr><td>2</td><td>0.20</td></tr> <tr><td>3</td><td>0.55</td></tr> <tr><td>4</td><td>-0.25</td></tr> <tr><td>5</td><td>-0.10</td></tr> </tbody> </table>

54794312

A residual plot for a linear regression model has residuals scattered around $0$ with no clear curve. One observation lies far to the right of all the others but has a residual very close to $0$. Consider the claim: “That observation cannot be influential because its residual is almost zero.” Explain why this conclusion does not follow from the residual plot alone.

Hints

- Separate vertical prediction error from an observation's horizontal location. - Ask what information a residual does and does not contain. - Consider whether one extreme explanatory-variable value could affect the fitted line itself.

Solution

1. A residual measures vertical prediction error, so a residual near $0$ only says that the fitted line predicts that observation well. 2. Because the observation is far from the others in the explanatory-variable direction, it has high leverage. 3. A high-leverage observation can strongly affect the fitted slope or intercept even when its residual is small. Influence must be assessed by examining how the fitted model changes when the observation is removed or by using an influence diagnostic.

Answer

The conclusion is not justified. The point has high leverage because its explanatory-variable value is extreme, and a high-leverage point can influence the fitted line even when its residual is near $0$.

54795112

Suppose residual is accidentally defined as predicted minus observed instead of observed minus predicted, and a residual plot is made using those incorrectly signed residuals. Describe how the incorrect residual plot is related to the correct residual plot. Which features of model adequacy would stay visible, and which interpretation would reverse?

Hints

- Compare the two residual definitions algebraically. - Ask what multiplying every vertical coordinate by the same negative constant does to a plot. - Separate pattern shape from the interpretation of the sign of a residual.

Solution

1. Reversing the residual definition multiplies every residual by $-1$. 2. The incorrect residual plot is the reflection of the correct residual plot across the horizontal line at $0$. 3. Curvature, changes in spread, and the locations of unusually large absolute residuals remain visible after reflection. 4. The sign interpretation reverses: a point shown above $0$ in the incorrect plot actually has a negative conventional residual, and vice versa.

Answer

The incorrect plot is a vertical reflection of the correct plot across residual $0$. Pattern shape and spread remain detectable, but the meanings of positive and negative residuals are reversed.

54804712

Suppose a “residual plot” is accidentally made using $|y-\hat{y}|$ on the vertical axis instead of the signed residual $y-\hat{y}$. Every plotted value is therefore at or above $0$. Consider the claim: “The model must systematically underpredict because all the residuals are positive.” Evaluate the claim and explain what information was lost by taking absolute values.

Hints

- Compare the quantity plotted with the definition of a residual. - Ask what the sign of a residual normally tells you. - Think about which features remain visible when every negative value is reflected above $0$.

Solution

1. The plotted quantities are absolute residuals, not signed residuals, so their nonnegative values do not indicate underprediction. 2. Taking absolute values removes whether each observation lies above or below the fitted line. 3. The plot can still show how prediction-error magnitude changes with $x$, but it cannot show sign-based patterns around $0$ in the usual residual-plot sense. 4. Therefore, the graph cannot support the claim that the model systematically underpredicts.

Answer

The claim is incorrect. Absolute residuals are always nonnegative, so they do not reveal whether the model overpredicts or underpredicts. Taking absolute values removes the residual signs and destroys part of the usual diagnostic information.

54809312

The residual plot comes from a regression of delivery time on driving distance. Open circles represent urban routes, and crosses represent highway routes. Consider the statement: “There is no curve, so the linear model has no systematic problem.” Evaluate the statement.

Hints

- Look for systematic structure in residuals beyond curves. - Compare where the two route types fall relative to residual $0$. - Interpret positive and negative residuals as underprediction and overprediction.

Solution

1. Lack of curvature addresses only whether a single straight-line form misses a curved relationship with distance. 2. Nearly all urban-route residuals are positive, while nearly all highway-route residuals are negative. 3. The model therefore tends to underpredict urban-route times and overpredict highway-route times. 4. This systematic separation suggests that route type contains information not represented by the one-predictor model, even though there is no curved pattern.

Answer

The statement is incorrect. The residuals show systematic bias by route type: urban routes are generally underpredicted and highway routes are generally overpredicted. No curvature does not mean that every important pattern is absent.

54810612

A fitted linear regression has slope $0$, so every fitted response equals $\bar{y}$. A residual plot using fitted values on the horizontal axis therefore shows only one vertical stack of residuals. Why is a residual plot against the original explanatory variable $x$ more informative in this situation?

Hints

- Determine what all fitted values are when the regression slope is zero. - Think about how many distinct horizontal positions remain in a residual-versus-fitted plot. - Ask which horizontal variable would preserve information about where each residual occurs in the original data.

Solution

1. With slope $0$, every fitted value is the same, so plotting residuals against fitted values gives every point the same horizontal coordinate. 2. That vertical stack cannot show whether residuals change systematically as $x$ changes. 3. Plotting the residuals against the original $x$-values preserves their horizontal ordering and can reveal curvature or another pattern that the zero-slope fit failed to capture.

Answer

Because all fitted values are identical, the residual-versus-fitted plot collapses horizontally. Plotting residuals against $x$ can still reveal systematic structure across the explanatory-variable range.

54812312

Two linear models are being evaluated. Model A has residuals spread roughly from $-8$ to $8$, with no visible pattern around zero. Model B has residuals mostly between $-3$ and $3$, but they form a clear curved pattern. Which residual plot gives stronger support for the linear form of the model? Explain without claiming that residual size is irrelevant.

Hints

- Evaluate residual pattern and residual magnitude as separate model qualities. - Ask what systematic curvature says about the chosen functional form. - Smaller errors do not automatically make the form of a model appropriate.

Solution

1. A random residual pattern around zero supports the assumption that a straight-line form is capturing the systematic relationship. 2. Model B's smaller residuals indicate smaller prediction errors on this scale, but the curvature shows systematic structure left unexplained by a straight line. 3. Therefore, Model A's residual plot gives stronger support for the linear form, although the larger residual spread still matters when evaluating predictive accuracy.

Answer

Model A gives stronger support for a linear form because its residuals show no systematic pattern. Model B has smaller residuals, but their curvature is evidence that a straight line is missing structure.

54812812

The residual plot comes from a calibration model. The three observations marked with crosses were collected on the same calibration day; the other observations were collected on several different days. What should be done with this information? Is it appropriate to delete the three crossed observations automatically?

Hints

- Compare the crossed observations with the rest of the residuals. - Identify what the unusual observations have in common besides their residual size. - Treat a diagnostic warning as a reason to investigate, not as proof that data should be removed.

Solution

1. The three crossed observations have unusually large positive residuals and share a common data-collection condition. 2. That pattern suggests investigating the calibration day for a recording, instrument, or procedural issue. 3. Large residuals alone do not prove that observations are erroneous, so automatic deletion would be unjustified. 4. The measurements should be verified, and any correction or exclusion should be documented and based on substantive evidence.

Answer

The shared calibration day should be investigated. The observations should not be deleted automatically; their large residuals flag them for investigation but do not by themselves show that the data are wrong.

54814012

Consider checking a regression by plotting each residual $e=y-\hat{y}$ against the corresponding observed response $y$. Why are residuals plotted against $x$ or against fitted values generally a cleaner diagnostic choice?

Hints

- Rewrite the observed response in terms of its fitted value and residual. - Check whether the proposed horizontal coordinate already contains the vertical coordinate. - Prefer a diagnostic that does not build the same error term directly into both axes.

Solution

1. The observed response contains the residual because $y=\hat{y}+e$. 2. Using $y$ on the horizontal axis can therefore create a built-in relationship between the horizontal coordinate and the residual, even when the model is otherwise appropriate. 3. Plotting residuals against $x$ or fitted values examines whether errors change systematically with the explanatory level or predicted response without placing the residual directly inside both axes.

Answer

Because $y=\hat{y}+e$, plotting $e$ against observed $y$ can create an artificial relationship. Residuals versus $x$ or fitted values are generally cleaner diagnostics for model structure.

54814912

The residual plot comes from a model whose response variable cannot be negative. What feature of the response scale may help explain the pattern near small fitted values, and why is caution warranted when using a simple linear-error model?

Hints

- Compare the residual positions at the left side of the plot with those farther right. - Use $y=\hat{y}+e$ to interpret a large negative residual near a fitted value of $0$. - Decide how a hard lower boundary affects the possible error distribution.

Solution

1. Near small fitted values, the residuals lie almost entirely above zero, while at moderate fitted values they extend on both sides of zero. 2. When the fitted response is close to the lower bound $0$, a large negative residual would imply an impossible negative observed response. 3. The lower boundary therefore limits how far residuals can extend below zero near small fitted values. 4. This asymmetry indicates that a constant, symmetric error model may not describe the response well near the boundary.

Answer

The lower bound at $0$ restricts negative residuals when fitted values are small. That boundary creates asymmetric errors, so a simple linear model with roughly symmetric constant error spread may be inappropriate near the lower end.

54814412

For an ordinary least-squares regression with an intercept, fit a second least-squares line with fitted values $\hat{y}$ as the explanatory variable and residuals $e$ as the response variable. What slope must this second line have, provided the fitted values are not all identical?

Hints

- Express the second regression slope using a centered cross-product in the numerator. - Relate fitted responses to the original least-squares line and its residuals. - Use the condition that the fitted values are not all identical when considering the denominator.

Solution

1. In an ordinary least-squares fit with an intercept, residuals are orthogonal to the fitted values, so $\sum (\hat{y}_i-\bar{\hat{y}})e_i=0$. 2. The slope from regressing residuals on fitted values is this centered cross-product sum divided by $\sum(\hat{y}_i-\bar{\hat{y}})^2$. 3. The numerator is $0$, and the denominator is positive because the fitted values are not all identical. 4. Therefore, the second least-squares line has slope $0$.

Answer

The second fitted line has slope $0$.

All problems may be used, copied and printed free of charge for school and tutoring, including paid tutoring. Commercial adaptations as well as publication or redistribution on the internet are not permitted.

Free math worksheets

Residual plots

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer

Hints

Solution

Answer