Residuals and Regression Diagnostics: Patterns and Influential Points

AP Statistics Topic Guide

Residuals and Regression Diagnostics: Patterns and Influential Points

Use residuals to measure prediction error, distinguish unusual y-values from leverage, and judge when a linear model needs closer scrutiny.

StatusCurrent regression support

Main keywordresiduals regression diagnostics

Worked cases12

Practice36 MCQs + 14 FRQs

Study progress0 completed

Residuals Regression Diagnostics: direct answer

Residuals regression diagnostics evaluate how observed responses differ from fitted values and whether unusual observations or systematic error patterns weaken a linear summary. Start with e=y−ŷ, then distinguish vertical outliers, leverage, influence, and extrapolation.

This page keeps the lesson centered on residuals regression diagnostics. Practice is included only after the method, assumptions, interpretation, and common decision points are explained.

Quick reference: Residuals and Regression Diagnostics: Patterns and Influential Points

Residual	e=y−ŷ
Residual plot	Look for random scatter around zero.

Concept mastery: Residuals and Regression Diagnostics: Patterns and Influential Points

Residuals are observed minus predicted

For an observation with response y and fitted value ŷ, the residual is e=y−ŷ. A positive residual means the observed response is above the fitted line and the model underpredicted that case; a negative residual means the model overpredicted. Residuals keep the units of the response variable, which is why a residual of 4 has no meaning until the response units are named.

The residual plot asks whether error is structureless

A useful linear model leaves residuals scattered around zero without a systematic curve or a steadily widening or narrowing band. The purpose is not to demand a perfectly random-looking picture. Instead, look for reproducible structure that the fitted line has failed to explain. A curved residual pattern warns that a straight-line summary is missing systematic form; a fan pattern warns that prediction variability changes across x.

Large residual and high leverage are different ideas

A response outlier has an unusual vertical departure from the fitted line, so its residual is large. A high-leverage point has an unusual explanatory-variable value, placing it far from the center of the x-values. One observation can have either property, both properties, or neither. High leverage creates the potential to rotate the fitted line; a large residual alone does not guarantee strong influence on slope.

Influence is assessed by change in the fitted model

An influential observation materially changes slope, intercept, correlation, fitted values, or another summary when it is removed or modified. Influence is therefore a comparative idea. Do not delete a point merely because it looks inconvenient. First check whether the value is a recording error, whether it belongs to the target population, and how conclusions change with and without it.

Prediction error and extrapolation require different checks

Residual analysis describes errors for observations represented by the fitted model. Extrapolation asks the line to predict outside the observed x-range, where the relationship may change entirely. A small residual standard pattern inside the observed range does not certify predictions far beyond that range. Context should control whether extrapolation is scientifically plausible.

Regression diagnostics support interpretation, not automatic model hunting

The revised AP Statistics course retains regression analysis while removing the older emphasis on analyzing departures from linearity as a separate topic. Use diagnostics to understand prediction error, unusual observations, and the reasonableness of a linear summary rather than turning the page into an advanced model-selection exercise.

12 worked residuals regression diagnostics cases

Worked case 1: Commute time

Scenario. A fitted model predicts a 31-minute commute for a worker whose observed commute is 38 minutes.

Reasoning. The residual is 38−31=7 minutes, so the point lies above the line and the model underpredicted by 7 minutes.

Case 1: Commute time check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 1; do not replace it with a memorized generic sentence.

Worked case 2: Plant growth

Scenario. A growth model predicts 18.4 cm while a measured plant reaches 16.1 cm.

Reasoning. The residual is −2.3 cm; the fitted line overpredicted this plant’s height.

Case 2: Plant growth check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 2; do not replace it with a memorized generic sentence.

Worked case 3: House price leverage

Scenario. Most homes are 900–2400 square feet, but one 6200-square-foot home is included.

Reasoning. The unusual x-value gives the large home high leverage even before its residual is examined.

Case 3: House price leverage check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 3; do not replace it with a memorized generic sentence.

Worked case 4: Test score outlier

Scenario. A student with an ordinary study-time value scores far below the fitted line.

Reasoning. The point can have a large negative residual without high leverage because its x-value is not unusual.

Case 4: Test score outlier check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 4; do not replace it with a memorized generic sentence.

Worked case 5: Curved residual pattern

Scenario. Residuals are positive for small x, negative in the middle, and positive again for large x.

Reasoning. The systematic curve indicates that a straight line leaves predictable structure unexplained.

Case 5: Curved residual pattern check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 5; do not replace it with a memorized generic sentence.

Worked case 6: Funnel residual pattern

Scenario. Residual spread grows steadily as predicted sales increase.

Reasoning. The model’s error variability is not constant across the fitted range, so high-sales predictions are less stable.

Case 6: Funnel residual pattern check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 6; do not replace it with a memorized generic sentence.

Worked case 7: Influential observation

Scenario. Deleting one high-leverage observation changes slope from 2.1 to 1.3.

Reasoning. The large change in fitted slope is direct evidence that the observation is influential.

Case 7: Influential observation check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 7; do not replace it with a memorized generic sentence.

Worked case 8: Interpolation

Scenario. Observed temperatures range from 10°C to 35°C and the model predicts at 24°C.

Reasoning. The prediction is interpolation because 24°C lies inside the observed explanatory-variable range.

Case 8: Interpolation check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 8; do not replace it with a memorized generic sentence.

Worked case 9: Extrapolation

Scenario. The same temperature model is used at −20°C.

Reasoning. The prediction is extrapolation and requires a contextual assumption that the linear relation continues far outside the observed range.

Case 9: Extrapolation check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 9; do not replace it with a memorized generic sentence.

Worked case 10: Zero residual

Scenario. A point lies exactly on the fitted line.

Reasoning. Its residual is 0, but that fact alone says nothing about whether it has high leverage.

Case 10: Zero residual check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 10; do not replace it with a memorized generic sentence.

Worked case 11: Residual units

Scenario. A regression predicts annual electricity use measured in kWh.

Reasoning. Every residual is measured in kWh because residuals are observed response minus predicted response.

Case 11: Residual units check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 11; do not replace it with a memorized generic sentence.

Worked case 12: Do not delete automatically

Scenario. A high-leverage point is verified as a genuine member of the target population.

Reasoning. Report its influence and sensitivity; unusual does not mean invalid, so deletion requires a scientific or data-quality reason.

Case 12: Do not delete automatically check: identify the exact statistical target in this scenario, verify the sign, denominator, assignment mechanism, event boundary, or model condition that controls the answer, and end with a conclusion whose scope matches the evidence. This check is specific to case 12; do not replace it with a memorized generic sentence.

Deep-dive notes for Residuals and Regression Diagnostics: Patterns and Influential Points

Residual size needs a scale

A residual of 5 can be negligible or severe depending on the response variable and the ordinary prediction error in the data. Five dollars in a house-price model may be trivial, while five percentage points in an election model can be large. Diagnose unusual residuals relative to the typical residual spread and the practical stakes, not from magnitude alone.

Standardized residuals are comparison tools

When residual spread varies or when observations need to be compared across fitted values, standardized residual ideas can help put deviations on a common scale. For AP-level interpretation, the essential point is simpler: a residual must be judged relative to ordinary model error. A raw vertical gap does not become influential merely because its number looks large.

Leverage comes from x, not y

A point can sit close to the fitted line and still have high leverage if its x-value is far from the rest. Such a point may strongly anchor the slope while showing a small residual. Conversely, a point near the center of the x-values can have a huge residual yet change the slope relatively little. Separate location in x from vertical error.

Influence is a sensitivity question

The clean diagnostic for influence is to compare results with and without the observation. If slope, intercept, fitted values, or correlation changes materially, the observation is influential for that summary. This sensitivity analysis does not automatically authorize deletion; it tells the analyst that conclusions depend strongly on one case and should be reported cautiously.

Residual plots preserve the x-order information

A list of residuals can show which observations were poorly predicted, but a residual plot reveals whether those errors follow x. Plot residuals against x or fitted values and look for systematic structure. Random scatter around zero supports a linear summary; visible organization means the model leaves predictable information behind.

Prediction claims need the observed range

A regression equation can always produce a numerical value, even at absurd x-values. Statistical justification is different. Check whether the requested x lies within the data range and whether the process can reasonably remain linear. A mechanically computed extrapolation should be labeled as such rather than presented with the confidence of an interpolation.

Unusual points can be scientifically important

Outliers and influential observations may represent data errors, but they may also reveal a subgroup, a regime change, or a mechanism the model does not capture. Verify the data and investigate context before removal. A model that fits only after discarding valid inconvenient cases may be hiding the most interesting part of the phenomenon.

R-squared does not replace diagnostics

A high coefficient of determination summarizes how much response variation is associated with the fitted linear model, but it does not certify linearity, constant spread, causation, or safe extrapolation. Residual behavior and study design still matter. Strong fit statistics and weak diagnostics can coexist, especially when leverage points dominate the regression.

Residuals and Regression Diagnostics: Patterns and Influential Points: 36 multiple-choice questions

These questions stay within this page’s topic. Work them after the concept and worked-case sections so practice reinforces the method rather than replacing instruction.

Question 1. Residuals and Regression Diagnostics

A regression model at a county election office in Capital Region during a school-year data collection is ŷ=6.092+2.261x. For x=20.172, the observed y is 52.383. Find and interpret the residual.

A. Residual 52.383 because the observed value is the error.
B. Residual 0.682; the model underpredicted by 0.682.
C. Residual 51.701 because the prediction is the unexplained component.
D. Residual -0.682; compute predicted minus observed.

Answer: B

Predicted y=6.092+2.261(20.172)=51.701. Residual=observed−predicted=52.383−51.701=0.682. The model underpredicted the observed response by 0.682 response units.

Question 2. Residuals and Regression Diagnostics

A regression model at a municipal water office in Mountain Region during a six-week field trial is ŷ=15.238+0.823x. For x=23.776, the observed y is 38.254. Find and interpret the residual.

A. Residual 34.806 because the prediction is the unexplained component.
B. Residual 3.448; the model underpredicted by 3.448.
C. Residual -3.448; compute predicted minus observed.
D. Residual 38.254 because the observed value is the error.

Answer: B

Predicted y=15.238+0.823(23.776)=34.806. Residual=observed−predicted=38.254−34.806=3.448. The model underpredicted the observed response by 3.448 response units.

Question 3. Residuals and Regression Diagnostics

A regression model at a regional airport authority in Midwest consortium during a quarterly performance study is ŷ=29.782+1.553x. For x=33.254, the observed y is 83.819. Find and interpret the residual.

A. Residual 83.819 because the observed value is the error.
B. Residual 81.425 because the prediction is the unexplained component.
C. Residual 2.394; the model underpredicted by 2.394.
D. Residual -2.394; compute predicted minus observed.

Answer: C

Predicted y=29.782+1.553(33.254)=81.425. Residual=observed−predicted=83.819−81.425=2.394. The model underpredicted the observed response by 2.394 response units.

Question 4. Residuals and Regression Diagnostics

A regression model at a recycling program in North Valley during a regional benchmarking study is ŷ=20.185+0.621x. For x=14.953, the observed y is 34.393. Find and interpret the residual.

A. Residual 34.393 because the observed value is the error.
B. Residual 29.471 because the prediction is the unexplained component.
C. Residual 4.922; the model underpredicted by 4.922.
D. Residual -4.922; compute predicted minus observed.

Answer: C

Predicted y=20.185+0.621(14.953)=29.471. Residual=observed−predicted=34.393−29.471=4.922. The model underpredicted the observed response by 4.922 response units.

Question 5. Residuals and Regression Diagnostics

A regression model at a public health department in Atlantic Corridor during a summer implementation review is ŷ=10.903+0.864x. For x=16.809, the observed y is 30.479. Find and interpret the residual.

A. Residual 5.053; the model underpredicted by 5.053.
B. Residual 30.479 because the observed value is the error.
C. Residual -5.053; compute predicted minus observed.
D. Residual 25.426 because the prediction is the unexplained component.

Answer: A

Predicted y=10.903+0.864(16.809)=25.426. Residual=observed−predicted=30.479−25.426=5.053. The model underpredicted the observed response by 5.053 response units.

Question 6. Residuals and Regression Diagnostics

A regression model at a community bank in Lakeside district during a follow-up evaluation period is ŷ=12.603+1.238x. For x=29.247, the observed y is 54.778. Find and interpret the residual.

A. Residual -5.967; compute predicted minus observed.
B. Residual 5.967; the model underpredicted by 5.967.
C. Residual 48.811 because the prediction is the unexplained component.
D. Residual 54.778 because the observed value is the error.

Answer: B

Predicted y=12.603+1.238(29.247)=48.811. Residual=observed−predicted=54.778−48.811=5.967. The model underpredicted the observed response by 5.967 response units.

Question 7. Residuals and Regression Diagnostics

A regression model at a university advising center in Atlantic Corridor during a six-week field trial is ŷ=20.757+2.011x. For x=15.31, the observed y is 55.961. Find and interpret the residual.

A. Residual -4.416; compute predicted minus observed.
B. Residual 51.545 because the prediction is the unexplained component.
C. Residual 4.416; the model underpredicted by 4.416.
D. Residual 55.961 because the observed value is the error.

Answer: C

Predicted y=20.757+2.011(15.31)=51.545. Residual=observed−predicted=55.961−51.545=4.416. The model underpredicted the observed response by 4.416 response units.

Question 8. Residuals and Regression Diagnostics

A regression model at a solar installer in Riverbend during a service-improvement study is ŷ=21.66+0.7x. For x=46.211, the observed y is 51.03. Find and interpret the residual.

A. Residual 2.978; compute predicted minus observed.
B. Residual 54.008 because the prediction is the unexplained component.
C. Residual -2.978; the model overpredicted by 2.978.
D. Residual 51.03 because the observed value is the error.

Answer: C

Predicted y=21.66+0.7(46.211)=54.008. Residual=observed−predicted=51.03−54.008=-2.978. The model overpredicted the observed response by 2.978 response units.

Question 9. Residuals and Regression Diagnostics

A regression model at a solar installer in Westview during a monthly quality review is ŷ=19.418+1.193x. For x=28.217, the observed y is 47.967. Find and interpret the residual.

A. Residual -5.114; the model overpredicted by 5.114.
B. Residual 5.114; compute predicted minus observed.
C. Residual 53.081 because the prediction is the unexplained component.
D. Residual 47.967 because the observed value is the error.

Answer: A

Predicted y=19.418+1.193(28.217)=53.081. Residual=observed−predicted=47.967−53.081=-5.114. The model overpredicted the observed response by 5.114 response units.

Question 10. Residuals and Regression Diagnostics

A regression model at a food safety laboratory in South Harbor during a multiweek validation study is ŷ=22.066+1.683x. For x=11.815, the observed y is 44.032. Find and interpret the residual.

A. Residual 2.081; the model underpredicted by 2.081.
B. Residual -2.081; compute predicted minus observed.
C. Residual 41.951 because the prediction is the unexplained component.
D. Residual 44.032 because the observed value is the error.

Answer: A

Predicted y=22.066+1.683(11.815)=41.951. Residual=observed−predicted=44.032−41.951=2.081. The model underpredicted the observed response by 2.081 response units.

Question 11. Residuals and Regression Diagnostics

A regression model at a city transit agency in Central County during a regional benchmarking study is ŷ=14.264+1.738x. For x=34.691, the observed y is 68.589. Find and interpret the residual.

A. Residual 74.557 because the prediction is the unexplained component.
B. Residual 5.968; compute predicted minus observed.
C. Residual -5.968; the model overpredicted by 5.968.
D. Residual 68.589 because the observed value is the error.

Answer: C

Predicted y=14.264+1.738(34.691)=74.557. Residual=observed−predicted=68.589−74.557=-5.968. The model overpredicted the observed response by 5.968 response units.

Question 12. Residuals and Regression Diagnostics

A regression model at a state park in Pine Ridge during a spring 2027 pilot is ŷ=5.751+1.759x. For x=37.771, the observed y is 76.157. Find and interpret the residual.

A. Residual 76.157 because the observed value is the error.
B. Residual 3.967; the model underpredicted by 3.967.
C. Residual 72.19 because the prediction is the unexplained component.
D. Residual -3.967; compute predicted minus observed.

Answer: B

Predicted y=5.751+1.759(37.771)=72.19. Residual=observed−predicted=76.157−72.19=3.967. The model underpredicted the observed response by 3.967 response units.

Question 13. Residuals and Regression Diagnostics

A regression model at a community bank in Pine Ridge during a spring 2027 pilot is ŷ=25.557+1.834x. For x=11.242, the observed y is 42.627. Find and interpret the residual.

A. Residual 3.548; compute predicted minus observed.
B. Residual 46.175 because the prediction is the unexplained component.
C. Residual 42.627 because the observed value is the error.
D. Residual -3.548; the model overpredicted by 3.548.

Answer: D

Predicted y=25.557+1.834(11.242)=46.175. Residual=observed−predicted=42.627−46.175=-3.548. The model overpredicted the observed response by 3.548 response units.

Question 14. Residuals and Regression Diagnostics

A regression model at a digital learning platform in Great Lakes during a randomized pilot period is ŷ=13.886+1.347x. For x=30.376, the observed y is 49.561. Find and interpret the residual.

A. Residual 54.802 because the prediction is the unexplained component.
B. Residual -5.241; the model overpredicted by 5.241.
C. Residual 49.561 because the observed value is the error.
D. Residual 5.241; compute predicted minus observed.

Answer: B

Predicted y=13.886+1.347(30.376)=54.802. Residual=observed−predicted=49.561−54.802=-5.241. The model overpredicted the observed response by 5.241 response units.

Question 15. Residuals and Regression Diagnostics

A regression model at a food safety laboratory in Sunbelt district during a randomized pilot period is ŷ=19.874+1.037x. For x=28.524, the observed y is 51.541. Find and interpret the residual.

A. Residual 49.453 because the prediction is the unexplained component.
B. Residual -2.088; compute predicted minus observed.
C. Residual 51.541 because the observed value is the error.
D. Residual 2.088; the model underpredicted by 2.088.

Answer: D

Predicted y=19.874+1.037(28.524)=49.453. Residual=observed−predicted=51.541−49.453=2.088. The model underpredicted the observed response by 2.088 response units.

Question 16. Residuals and Regression Diagnostics

A regression model at a farm cooperative in Capital Region during a summer implementation review is ŷ=13.863+0.679x. For x=18.176, the observed y is 31.145. Find and interpret the residual.

A. Residual 26.205 because the prediction is the unexplained component.
B. Residual 31.145 because the observed value is the error.
C. Residual -4.94; compute predicted minus observed.
D. Residual 4.94; the model underpredicted by 4.94.

Answer: D

Predicted y=13.863+0.679(18.176)=26.205. Residual=observed−predicted=31.145−26.205=4.94. The model underpredicted the observed response by 4.94 response units.

Question 17. Residuals and Regression Diagnostics

A regression model at a university advising center in North Valley during a semester-long cohort study is ŷ=29.331+2.401x. For x=44.308, the observed y is 137.4. Find and interpret the residual.

A. Residual 137.4 because the observed value is the error.
B. Residual 135.7 because the prediction is the unexplained component.
C. Residual 1.678; the model underpredicted by 1.678.
D. Residual -1.678; compute predicted minus observed.

Answer: C

Predicted y=29.331+2.401(44.308)=135.7. Residual=observed−predicted=137.4−135.7=1.678. The model underpredicted the observed response by 1.678 response units.

Question 18. Residuals and Regression Diagnostics

A regression model at a housing authority in Sunbelt district during a two-month observation window is ŷ=14.797+2.278x. For x=36.39, the observed y is 92.483. Find and interpret the residual.

A. Residual 5.21; compute predicted minus observed.
B. Residual -5.21; the model overpredicted by 5.21.
C. Residual 92.483 because the observed value is the error.
D. Residual 97.693 because the prediction is the unexplained component.

Answer: B

Predicted y=14.797+2.278(36.39)=97.693. Residual=observed−predicted=92.483−97.693=-5.21. The model overpredicted the observed response by 5.21 response units.

Question 19. Residuals and Regression Diagnostics

A regression model at a municipal emergency dispatch center in Westview during a community outreach cycle is ŷ=17.808+1.282x. For x=40.285, the observed y is 72.731. Find and interpret the residual.

A. Residual 3.278; the model underpredicted by 3.278.
B. Residual 72.731 because the observed value is the error.
C. Residual -3.278; compute predicted minus observed.
D. Residual 69.453 because the prediction is the unexplained component.

Answer: A

Predicted y=17.808+1.282(40.285)=69.453. Residual=observed−predicted=72.731−69.453=3.278. The model underpredicted the observed response by 3.278 response units.

Question 20. Residuals and Regression Diagnostics

A regression model at a regional airport authority in Pacific Northwest during a fall 2026 audit is ŷ=7.432+0.888x. For x=42.483, the observed y is 45.646. Find and interpret the residual.

A. Residual 45.157 because the prediction is the unexplained component.
B. Residual 0.489; the model underpredicted by 0.489.
C. Residual 45.646 because the observed value is the error.
D. Residual -0.489; compute predicted minus observed.

Answer: B

Predicted y=7.432+0.888(42.483)=45.157. Residual=observed−predicted=45.646−45.157=0.489. The model underpredicted the observed response by 0.489 response units.

Question 21. Residuals and Regression Diagnostics

A regression model at a city transit agency in North Valley during a multiweek validation study is ŷ=13.784+0.552x. For x=28.434, the observed y is 27.75. Find and interpret the residual.

A. Residual -1.73; the model overpredicted by 1.73.
B. Residual 27.75 because the observed value is the error.
C. Residual 29.48 because the prediction is the unexplained component.
D. Residual 1.73; compute predicted minus observed.

Answer: A

Predicted y=13.784+0.552(28.434)=29.48. Residual=observed−predicted=27.75−29.48=-1.73. The model overpredicted the observed response by 1.73 response units.

Question 22. Residuals and Regression Diagnostics

A regression model at a city recreation department in Metro East during a service-improvement study is ŷ=28.063+1.01x. For x=31.906, the observed y is 56.42. Find and interpret the residual.

A. Residual 56.42 because the observed value is the error.
B. Residual 60.288 because the prediction is the unexplained component.
C. Residual 3.868; compute predicted minus observed.
D. Residual -3.868; the model overpredicted by 3.868.

Answer: D

Predicted y=28.063+1.01(31.906)=60.288. Residual=observed−predicted=56.42−60.288=-3.868. The model overpredicted the observed response by 3.868 response units.

Question 23. Residuals and Regression Diagnostics

A regression model at a county library in Central County during a fall 2026 audit is ŷ=27.723+1.972x. For x=40.158, the observed y is 107.5. Find and interpret the residual.

A. Residual -0.573; compute predicted minus observed.
B. Residual 107.5 because the observed value is the error.
C. Residual 0.573; the model underpredicted by 0.573.
D. Residual 106.9 because the prediction is the unexplained component.

Answer: C

Predicted y=27.723+1.972(40.158)=106.9. Residual=observed−predicted=107.5−106.9=0.573. The model underpredicted the observed response by 0.573 response units.

Question 24. Residuals and Regression Diagnostics

A regression model at a food safety laboratory in Central County during a winter readiness review is ŷ=17.857+1.227x. For x=17.245, the observed y is 39.852. Find and interpret the residual.

A. Residual 0.835; the model underpredicted by 0.835.
B. Residual 39.017 because the prediction is the unexplained component.
C. Residual -0.835; compute predicted minus observed.
D. Residual 39.852 because the observed value is the error.

Answer: A

Predicted y=17.857+1.227(17.245)=39.017. Residual=observed−predicted=39.852−39.017=0.835. The model underpredicted the observed response by 0.835 response units.

Question 25. Residuals and Regression Diagnostics

A regression model at a public health department in Great Lakes during a winter readiness review is ŷ=26.094+1.95x. For x=26.433, the observed y is 84.542. Find and interpret the residual.

A. Residual -6.904; compute predicted minus observed.
B. Residual 77.638 because the prediction is the unexplained component.
C. Residual 84.542 because the observed value is the error.
D. Residual 6.904; the model underpredicted by 6.904.

Answer: D

Predicted y=26.094+1.95(26.433)=77.638. Residual=observed−predicted=84.542−77.638=6.904. The model underpredicted the observed response by 6.904 response units.

Question 26. Residuals and Regression Diagnostics

A regression model at a housing authority in Sunbelt district during a six-week field trial is ŷ=28.845+1.723x. For x=20.141, the observed y is 70.654. Find and interpret the residual.

A. Residual -7.106; compute predicted minus observed.
B. Residual 63.548 because the prediction is the unexplained component.
C. Residual 70.654 because the observed value is the error.
D. Residual 7.106; the model underpredicted by 7.106.

Answer: D

Predicted y=28.845+1.723(20.141)=63.548. Residual=observed−predicted=70.654−63.548=7.106. The model underpredicted the observed response by 7.106 response units.

Question 27. Residuals and Regression Diagnostics

A regression model at a regional airport authority in North Valley during a regional benchmarking study is ŷ=22.38+1.132x. For x=26.627, the observed y is 55.583. Find and interpret the residual.

A. Residual 3.061; the model underpredicted by 3.061.
B. Residual 52.522 because the prediction is the unexplained component.
C. Residual -3.061; compute predicted minus observed.
D. Residual 55.583 because the observed value is the error.

Answer: A

Predicted y=22.38+1.132(26.627)=52.522. Residual=observed−predicted=55.583−52.522=3.061. The model underpredicted the observed response by 3.061 response units.

Question 28. Residuals and Regression Diagnostics

A regression model at a solar installer in Pacific Northwest during a multiweek validation study is ŷ=12.089+1.174x. For x=26.598, the observed y is 49.626. Find and interpret the residual.

A. Residual 43.315 because the prediction is the unexplained component.
B. Residual -6.311; compute predicted minus observed.
C. Residual 49.626 because the observed value is the error.
D. Residual 6.311; the model underpredicted by 6.311.

Answer: D

Predicted y=12.089+1.174(26.598)=43.315. Residual=observed−predicted=49.626−43.315=6.311. The model underpredicted the observed response by 6.311 response units.

Question 29. Residuals and Regression Diagnostics

A regression model at a regional hospital in North Valley during a spring 2027 pilot is ŷ=20.617+1.846x. For x=21.39, the observed y is 66.129. Find and interpret the residual.

A. Residual 60.103 because the prediction is the unexplained component.
B. Residual 66.129 because the observed value is the error.
C. Residual -6.026; compute predicted minus observed.
D. Residual 6.026; the model underpredicted by 6.026.

Answer: D

Predicted y=20.617+1.846(21.39)=60.103. Residual=observed−predicted=66.129−60.103=6.026. The model underpredicted the observed response by 6.026 response units.

Question 30. Residuals and Regression Diagnostics

A regression model at a housing authority in Metro East during a fall 2026 audit is ŷ=21.627+0.631x. For x=32.273, the observed y is 38.486. Find and interpret the residual.

A. Residual 3.505; compute predicted minus observed.
B. Residual -3.505; the model overpredicted by 3.505.
C. Residual 41.991 because the prediction is the unexplained component.
D. Residual 38.486 because the observed value is the error.

Answer: B

Predicted y=21.627+0.631(32.273)=41.991. Residual=observed−predicted=38.486−41.991=-3.505. The model overpredicted the observed response by 3.505 response units.

Question 31. Residuals and Regression Diagnostics

A regression model at a public high school in Riverbend during a spring 2027 pilot is ŷ=18.382+2.046x. For x=40.441, the observed y is 96.697. Find and interpret the residual.

A. Residual 101.1 because the prediction is the unexplained component.
B. Residual -4.427; the model overpredicted by 4.427.
C. Residual 96.697 because the observed value is the error.
D. Residual 4.427; compute predicted minus observed.

Answer: B

Predicted y=18.382+2.046(40.441)=101.1. Residual=observed−predicted=96.697−101.1=-4.427. The model overpredicted the observed response by 4.427 response units.

Question 32. Residuals and Regression Diagnostics

A regression model at a recycling program in New England network during a regional benchmarking study is ŷ=21.384+1.564x. For x=26.841, the observed y is 66.418. Find and interpret the residual.

A. Residual 66.418 because the observed value is the error.
B. Residual 63.363 because the prediction is the unexplained component.
C. Residual -3.055; compute predicted minus observed.
D. Residual 3.055; the model underpredicted by 3.055.

Answer: D

Predicted y=21.384+1.564(26.841)=63.363. Residual=observed−predicted=66.418−63.363=3.055. The model underpredicted the observed response by 3.055 response units.

Question 33. Residuals and Regression Diagnostics

A regression model at a solar installer in Capital Region during a pre-exam training cycle is ŷ=9.817+2.242x. For x=23.899, the observed y is 60.549. Find and interpret the residual.

A. Residual -2.85; the model overpredicted by 2.85.
B. Residual 2.85; compute predicted minus observed.
C. Residual 60.549 because the observed value is the error.
D. Residual 63.399 because the prediction is the unexplained component.

Answer: A

Predicted y=9.817+2.242(23.899)=63.399. Residual=observed−predicted=60.549−63.399=-2.85. The model overpredicted the observed response by 2.85 response units.

Question 34. Residuals and Regression Diagnostics

A regression model at a wildlife clinic in Lakeside district during a follow-up evaluation period is ŷ=16.579+0.943x. For x=20.956, the observed y is 41.983. Find and interpret the residual.

A. Residual -5.642; compute predicted minus observed.
B. Residual 36.341 because the prediction is the unexplained component.
C. Residual 41.983 because the observed value is the error.
D. Residual 5.642; the model underpredicted by 5.642.

Answer: D

Predicted y=16.579+0.943(20.956)=36.341. Residual=observed−predicted=41.983−36.341=5.642. The model underpredicted the observed response by 5.642 response units.

Question 35. Residuals and Regression Diagnostics

A regression model at a food safety laboratory in Riverbend during a quarterly performance study is ŷ=28.21+1.567x. For x=10.357, the observed y is 38.508. Find and interpret the residual.

A. Residual 38.508 because the observed value is the error.
B. Residual 5.931; compute predicted minus observed.
C. Residual 44.439 because the prediction is the unexplained component.
D. Residual -5.931; the model overpredicted by 5.931.

Answer: D

Predicted y=28.21+1.567(10.357)=44.439. Residual=observed−predicted=38.508−44.439=-5.931. The model overpredicted the observed response by 5.931 response units.

Question 36. Residuals and Regression Diagnostics

A regression model at a city recreation department in Cedar Grove during a six-week field trial is ŷ=7.191+2.308x. For x=26.572, the observed y is 63.35. Find and interpret the residual.

A. Residual -5.169; the model overpredicted by 5.169.
B. Residual 68.519 because the prediction is the unexplained component.
C. Residual 63.35 because the observed value is the error.
D. Residual 5.169; compute predicted minus observed.

Answer: A

Predicted y=7.191+2.308(26.572)=68.519. Residual=observed−predicted=63.35−68.519=-5.169. The model overpredicted the observed response by 5.169 response units.

Residuals and Regression Diagnostics: Patterns and Influential Points: 14 free-response questions

For each response, show the statistical reasoning, use the scenario’s language, and state only the conclusion supported by the design or probability model.

FRQ set 1: Residuals and Regression Diagnostics

Scenario. A regression model at a municipal water office in Mountain Region during a winter readiness review is ŷ=8.018+0.652x. For x=38.799, the observed y is 34.781. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=8.018+0.652(38.799)=33.315. Residual=observed−predicted=34.781−33.315=1.466. The model underpredicted the observed response by 1.466 response units.

FRQ set 2: Residuals and Regression Diagnostics

Scenario. A regression model at a city recreation department in New England network during a service-improvement study is ŷ=12.569+0.804x. For x=11.629, the observed y is 18.204. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=12.569+0.804(11.629)=21.919. Residual=observed−predicted=18.204−21.919=-3.715. The model overpredicted the observed response by 3.715 response units.

FRQ set 3: Residuals and Regression Diagnostics

Scenario. A regression model at a county election office in Lakeside district during a winter readiness review is ŷ=14.121+1.001x. For x=13.789, the observed y is 24.678. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=14.121+1.001(13.789)=27.924. Residual=observed−predicted=24.678−27.924=-3.246. The model overpredicted the observed response by 3.246 response units.

FRQ set 4: Residuals and Regression Diagnostics

Scenario. A regression model at a farm cooperative in Westview during a winter readiness review is ŷ=28.388+1.314x. For x=24.097, the observed y is 52.446. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=28.388+1.314(24.097)=60.051. Residual=observed−predicted=52.446−60.051=-7.605. The model overpredicted the observed response by 7.605 response units.

FRQ set 5: Residuals and Regression Diagnostics

Scenario. A regression model at a regional manufacturer in Pacific Northwest during a monthly quality review is ŷ=26.373+1.518x. For x=18.974, the observed y is 53.383. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=26.373+1.518(18.974)=55.176. Residual=observed−predicted=53.383−55.176=-1.793. The model overpredicted the observed response by 1.793 response units.

FRQ set 6: Residuals and Regression Diagnostics

Scenario. A regression model at a county library in Great Lakes during a fall 2026 audit is ŷ=6.524+1.955x. For x=49.362, the observed y is 95.165. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=6.524+1.955(49.362)=103.0. Residual=observed−predicted=95.165−103.0=-7.862. The model overpredicted the observed response by 7.862 response units.

FRQ set 7: Residuals and Regression Diagnostics

Scenario. A regression model at a city recreation department in South Harbor during a semester-long cohort study is ŷ=18.518+0.888x. For x=25.312, the observed y is 36.023. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=18.518+0.888(25.312)=40.995. Residual=observed−predicted=36.023−40.995=-4.972. The model overpredicted the observed response by 4.972 response units.

FRQ set 8: Residuals and Regression Diagnostics

Scenario. A regression model at a community college in Pine Ridge during a summer implementation review is ŷ=6.569+0.548x. For x=38.007, the observed y is 30.906. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=6.569+0.548(38.007)=27.397. Residual=observed−predicted=30.906−27.397=3.509. The model underpredicted the observed response by 3.509 response units.

FRQ set 9: Residuals and Regression Diagnostics

Scenario. A regression model at a regional airport authority in Pacific Northwest during a fall 2026 audit is ŷ=6.628+1.207x. For x=47.804, the observed y is 64.491. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=6.628+1.207(47.804)=64.327. Residual=observed−predicted=64.491−64.327=0.164. The model underpredicted the observed response by 0.164 response units.

FRQ set 10: Residuals and Regression Diagnostics

Scenario. A regression model at a county election office in Lakeside district during a baseline measurement week is ŷ=6.141+1.678x. For x=28.58, the observed y is 46.248. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=6.141+1.678(28.58)=54.098. Residual=observed−predicted=46.248−54.098=-7.85. The model overpredicted the observed response by 7.85 response units.

FRQ set 11: Residuals and Regression Diagnostics

Scenario. A regression model at a university advising center in Midwest consortium during a school-year data collection is ŷ=17.983+2.032x. For x=16.256, the observed y is 58.905. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=17.983+2.032(16.256)=51.015. Residual=observed−predicted=58.905−51.015=7.89. The model underpredicted the observed response by 7.89 response units.

FRQ set 12: Residuals and Regression Diagnostics

Scenario. A regression model at a farm cooperative in North Valley during a follow-up evaluation period is ŷ=8.234+2.356x. For x=36.204, the observed y is 86.929. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=8.234+2.356(36.204)=93.531. Residual=observed−predicted=86.929−93.531=-6.602. The model overpredicted the observed response by 6.602 response units.

FRQ set 13: Residuals and Regression Diagnostics

Scenario. A regression model at a public high school in Westview during a regional benchmarking study is ŷ=7.227+2.329x. For x=30.656, the observed y is 82.687. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=7.227+2.329(30.656)=78.625. Residual=observed−predicted=82.687−78.625=4.062. The model underpredicted the observed response by 4.062 response units.

FRQ set 14: Residuals and Regression Diagnostics

Scenario. A regression model at a community bank in Pacific Northwest during a monthly quality review is ŷ=14.345+0.699x. For x=37.434, the observed y is 46.14. Find and interpret the residual.

Define the variable, units, groups, and requested distribution or model feature.
Show the required calculation or graphical/model reasoning with labeled quantities.
Interpret the numerical result in the context of the data rather than as an isolated number.
Identify an unusual feature, limitation, or condition that affects the conclusion.

Model response

Predicted y=14.345+0.699(37.434)=40.511. Residual=observed−predicted=46.14−40.511=5.629. The model underpredicted the observed response by 5.629 response units.

Continue with a connected AP Statistics skill

Use the next page to connect this method to a neighboring concept rather than repeating the same calculation pattern.